Course Blog Mathematical Economics

Synopses: Tutorials 3, 4, and 5 (Ac. Year 2025-26).

Mon, 11 May 2026 00:50:39 +0300

Dear Students,

Below you can find a synopsis of our last three tutorials, as well as links for material that we covered in class.

In the third tutorial, we discussed in depth the notion of total boundedness and its connection with boundedness. We solved exercises 11 and 12 from Problem Set 2. Regarding total boundedness, you are advised to study exercise 8 from the same problem set.

In the fourth tutorial, we proved that when a certain functional inequality between two metrics hold, then the convergence and the Cauchy property is inherited from one metric to the other. We also discussed how we should strengthen the functional inequality in order to get the preservation of completeness between the two metrics.

Finally, we discussed and proved an important result in mathematical analysis, according to which the set of bounded functions from a set X to a metric space Y is complete w.r.t. the unifrom metric if the codomain space Y is complete. Relevant notes on this result can be found here. This material (the version that we followed in class) can also be found here.

Finally, in the fifth tutorial we discussed and proved some basic results about the principle of optimality. We motivated our analysis by connecting the ideas behind the principle of optimality with a standard example from growth theory. Afterwards, we stated a general dynamic programming problem, and its corresponding Bellman equation, and we provided brief proofs about the (conditional) equivalence of their solution sets. In the class we followed these notes, and we covered material until the end of Lemma 1, as well as Theorems 3 and 4. You are also advised to take a look at the notes by Dr. Zaverdas (they contain essentially the same proofs but with different notation), and to study the numerical example from page 9 onwards.

Please do not hesitate to reach out for any questions.

Thanks

Pantelis

Synopsis: Lecture 12 (Ac. Year 2025-26)

Sun, 03 May 2026 04:01:32 +0300

We proceeded with the formulation and derivation of the Banach Fixed Point Theorem and some corollaries involving localization of the fixed point via the restriction into closed and invariant subspaces, as well as regarding the approximation error of the fixed point by elements of the sequence of iterations. We have started our preparations for the examination of applications involving the establishement of the existence and uniqueness of functional equations.

Notes for the above can be found here and here. The whiteboards from analogous 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here and here .

Exercise: Using the BFPT and the result in Optional Exercise 11, show that if the underlying metric space is totally bounded and complete, and the f function is contractive, then its m-fold self-composition converges uniformly (w.r.t. which metric?) to the function that is constant at the unique fixed point of f.

Synopsis: Lecture 11 (Ac. Year 2025-26)

Sat, 25 Apr 2026 19:52:44 +0300

In preparation for the development of the Banach fixed point theorem, we have proceeded with the examination of the notion Lipschitz continuity. a (along with some work that involved for example the characterization of Lipschitz continuity for functions between Euclidean spaces-see here for the develpment of the full details) and contraction mappings (along with self-composition properties). We proceeded with the introduction of some general aspects of fixed point theory.

Notes for the above can be found here. The whiteboards from analogous Ac. year's 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, and here.

Synopsis Lecture 10+1/2 (Ac. Year 2025-26)

Tue, 07 Apr 2026 16:14:36 +0300

We have shown that pointwise convergence is strictly weaker than uniform, and we have provided a theorem that complements pointwise convergence with a joint Lipschitz continuity property for the elements of the sequence, and total boundedness of the common domain in order to obtain uniform convergence.

We have examined the usefulness of the strictness of uniform convergence in a general application that concerns the issue approximation of optimization problems (variational approximation). We have proven that uniform convergence implies convergence of optimal values, and by equipping the underlying domain with a metric, w.r.t. which the limit function has a well-distinguishable optimizer, we have extended the convergence to the optimizers.

We have proceeded with the brief examination of the notion of completeness.

Notes for the above can be found here, and here. The whiteboards from analogous Ac. year's 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, and here.

Exercises:

The examination of variational approximation was indicatively performed for problems of maximization. Derive the analogous results for minimization.
Try to deduce whether pointwise convergence suffices for the approximation of maxima of strictly concave functions. (not trivial!)

Synopsis: Lecture 9 (Ac. Year 2025-26)

Sun, 29 Mar 2026 22:42:33 +0300

We formulated an example of application of the ULLN using an iid standard uniform process along with a function space defined on the relevant support that we have already established to be totally bounded wrt the uniform metric. We claimed that the above could be useful in facilitating the convergence in probability of a stochastic minimizer to a limiting deterministic analogue, thereby providing a mathematical structure relevant for the establishment of weak consistency of econometric estimators.

In order to make sense of the above in our metric space language, we provided a brief reminder of topological notions in metric spaces with emphasis on the notion of (sequential) convergence and function continuity. Specifying the above in functional speces equipped with the uniform metric we have focused on the notion of uniform functional convergence constrasting it to the (generically non metrizable) notion of pointwise convergence. We will be using this in order to focus on a major application: the approximation of optimization problems.

Notes for the above can be found here and here. The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Synopsis: Lecture 8 (Ac. Year 2025-26)

Sun, 22 Mar 2026 00:14:26 +0300

Given our parametric example, we have proven that a function space endowed with the uniform metric, consisting of functions that are bijectively parameterized by the elements of a totally bounded metric space, and if the parameterization obeys a Lipschitz continuity property, is totally bounded.

We have continued with our application of the notion of total boundedness in asymptotic analysis: the derivation of a Uniform Law of Large Numbers in a simple framework using a chaining argument that makes use of finite coverings. We have constructed examples of totally bounded sets via the notion of covering numbers.

Notes on the above can be found here. Notes on the particular ULLN application can be found here.

The whiteboards from analogous lectures from the Av. Year 2020-21 (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Synopsis: Lectures 6 and 7 (Ac. Year 2025-26)

Sun, 15 Mar 2026 13:54:52 +0300

We have initiated our examination of the notion of total boundness. We have given properties, e.g. that the centers of the covering balls can be chosen inside the totally bounded set, as well as (trivial) examples-counterexamples (the analytical complexity of the notion clearly manifested itself on that counter examples were easier to come up to) using among others the property of sequential disconnectedness.

Given the analytical complexity of the verification of total boundness, we will use the introduction of the notions of covering numbers and metric entropy in order to construct examples and further properties. We have occupied ourselves with application relevant examples in function spaces.

Notes on the above can be found here, and here.

The whiteboards from analogous academic year's 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, and here.

Synopses: Tutorials 1 and 2 (Ac. Year 2025-26)

Mon, 09 Mar 2026 23:50:50 +0300

Dear Students,

In the first tutorial, we stated the definitions of a metric and a pseudo-metric and we solved exercises 4.8 and 5 from Problem Set 1 (it can be found in the folder TA Sessions 2023-2024 -> TA Session 1). We also presented an example of a metric reflecting the structure of a graph. Specifically, if the graph is connected, the metric is well-defined as the minimum distance between any two vertices of the graph. The relevant notes are here.

In the second tutorial we discussed the definition of a bounded function from a non-empty set X to a metric space Y, and we showed that the set of such functions, B(X,Y), together with the uniform metric, is a well-defined metric space. Moreover, we proved that the boundedness of the metric space Y is inherited to the functional space B(X,Y) (exercise 7, TA Sessions 2023-2024, Problem Set 2). From the same Problem Set, we solved exercises 3, 4, and 6. We also noted that in any metric space (X,d), the boundedness of a subset A, is equivalent to the condition: d(x,y) <= e, for some e>0, for all x,y in A.

You are advised to try exercises 6, 7 and 8 from Problem Set 1, as well as exercises 1, 2, and 5 from Problem Set 2.

Feel free to reach out for any questions.

Pantelis

Synopses: Lectures 4 and 5+1/2 (Ac. Year 2025-26)

Sun, 22 Feb 2026 01:35:15 +0300

We continued studying properties of metric induced balls. We have shown that these obey some monotonicity property. The examples examined including the ones of the real line endowed with the usual metric, the real line endowed with the "exponential metric", the real line endowed with the discrete one, showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric.

We have provided with the geometric realizations of open and closed balls in endowed with any of the three "commonly examined" metrics, showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric. Their geometric relations were later on exemplified via the notion of metric equivalence and its balls' implications. We thus examined the antimonotonic relationship between balls of the same centered arising by pairs of dominant and dominating metrics. We have used the norions of cylinders in order to obtain geometric depictions of balls wrt the uniform metric in convenient domains.

We have begun to show several general poperties of the notion and provide with examples: e.g. open and closed balls are by construction bounded. The notion is hereditary. If a set is bounded then any subset is also bounded. The dual is evident. If a set is not bounded then any superset is also unbounded. Obviously the notion depends crucially on the metric. For example the discrete space is always bounded. This is in contrast with any Euclidean space (i.e. the equipped with the usual metric) which is not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one.

We have proven that the center of the covering ball need not be an element of the subset at hand although we have shown that when such a ball exists the center can always be chosen to lie inside this subset (or anywhere inside the space as long as the radius id appropriately modified). We have shown that any finite set is (universally) bounded. In function spaces we have examined the notion of uniform boundedness (function boundedness via a common bound) via a variational property and found it to be conveniently equivalent to boundedness w.r.t. the uniform metric.

We examined the relation of boundedness with the concepts of metric dominance and equivalent and proven that equivalent metrics (even though generically different) completely agree on the sets they consider bounded.

Finally, using the notion of finite open (closed) covers, we formed an equivalent definition of boundedness that in turn motivated its strenthening to the notion of total boundedness.

Notes on the above can be found here, here, here, and here.

Whiteboards from previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.

Synopsis: Lecture 3 (Ac. Year 2025-26)

Sat, 14 Feb 2026 23:55:53 +0300

Given relations we have established in our real vector spaces examples, we have defined the notions of dominance and equivalence between metrics definable on the same carrier set. Two such metrics can be considered equivalent-and we suspected that the notion is representing "similar properties" on the same carrier-without being necessarily equal as functions.

We examined the important-to the upcoming lectures-example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and noted that it contains several subexamples, e.g. the spaces of real n-vectors equipped with the max-metric.

Studying the example of the n-dimensional Hamming space, we introduced the useful notion of a metric sub-space.

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty.

Notes on the previous can be found here and here. The whiteboards from analogous previous year's lectures (please keep in mind that those arenot necessarily identical to the current lectures but they contain some common elements) can be found here and here.

Exercise

Show that for some appropriate .

Synopsis: Lecture 2 (Ac. Year 2025-26)

Sun, 08 Feb 2026 20:14:21 +0300

We continued the investigation of examples involving metrics on sets of finite dimensional real vectors. (Some of) The examples have shown that it is possible that different metrics on the same carrier set can obey relations, e.g. in the form of functional inequalities. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as with a motivation of further examination of such relations.

Notes on the previous can be found here. The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Exercise. Suppose that , , and consider . Show that a well-defined metric? What if the weights? Would a specific choice of the weights give one of our examined metrics?

Synopsis: Lecture 1 (Ac. Year 2025-26)

Sun, 01 Feb 2026 20:10:13 +0300

After a presentation of the course's scope and aims, and using the overview of the familiar case of the real numbers, we begun with the definition of a distance function (metric) over a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families of metrics, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space.

Procedural details can be found at the course's syllabus, notes on the previous here, and a counterexample of a metric here. The whiteboards from the analogous lecture of the Ac. Year 2020-21 (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Exercise: Using as carrier set the real line, provide a counterexample of a function that fails to be a metric because it fails to satisfy the triangle inequality, even though it satisfies the remaining properties of the definition.

Exercise: Show that the discrete metric is actually an ultrametric.

Lectures 15-16 (include the 3nd complementary lecture-Ac. Year 2023-24)

Sat, 18 May 2024 17:28:50 +0300

After a brief remark on the characterization of Lipschitz continuity for functions between Euclidean spaces (see here for the develpment of the full details), we proceeded with the introduction of some general aspects of fixed point theory, and the examination of the proof and corollaries of the Banach Fixed Point Theorem. We have started our preparations for the examination of applications involving the establishement of the existence and uniqueness of functional equations, by among others proving Blackwell's Lemma. We concluded the course with an applications of the BFPT; it involves the uniqueness of solution to the Bellman equation.

Lectures 13-14 (include the 2nd complementary lecture-Ac. Year 2023-24)

Sat, 27 Apr 2024 17:30:20 +0300

We have completed the relation of uniform convergence with the approximation of optimization problems by focusing on the asymptotic behavior of optimizers. We have proceeded with the examination of the notions of completeness and Lipschitz continuity, and contraction mappings.

Notes for the above can be found here, here and here. The whiteboards from analogous Ac. year's 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, here, and here.

Lectures 11-12 (include the first complementary lecture-Ac.Year 2023-24)

Sun, 21 Apr 2024 23:48:35 +0300

We provided a brief reminder of topological notions in metric spaces with emphasis on the notion of (sequential) convergence and function continuity. Specifying the above in functional speces equipped with the uniform metric we have focused on the notion of uniform functional convergence constrasting it to the (generically non metrizable) notion of pointwise convergence. We then focused on a major application: the approximation of optimization problems.

Lectures 9-10 (Ac. Year 2023-24)

Sat, 06 Apr 2024 15:59:51 +0300

Notes on the above can be found here. Notes on the particular ULLN application can be found here.

Lecture 8 (Ac. Year 2023-24)

Sun, 31 Mar 2024 18:31:08 +0300

We begun with an equivalent definition that is useful for the introduction of total boundness. Strenghtening thus, the boundedness characterization we have introduced the notion of total boundness. We have started our examination of the latter. We have given properties, e.g. that the centers of the covering balls can be chosen inside the totally bounded set, as well as (trivial) examples-counterexamples (the analytical complexity of the notion clearly manifested itself on that counter examples were easier to come up to).

We have instead begun the investigation of an application of the notion of total boundedness in asymptotic analysis, by deriving a Uniform Law of Large Numbers in a simple framework.

Notes on the above can be found here, and here. Notes on the ULLN application can be found here.

Lecture 7 (Ac. Year 2023-24)

Sat, 23 Mar 2024 19:05:29 +0300

We continued our study of the notion of boundedness. We have proven that the center of the covering ball need not be an element of the subset at hand although we have shown that when such a ball exists the center can always be chosen to lie inside this subset. We have shown that any finite set is (universally) bounded. We have proven that if a space is bounded w.r.t. a dominant metric, then it is also bounded w.r.t. to the dominated one. An obvious corollary is that equivalent metrics totally agree on what parts of the metric space are considered bounded.

We have provided with further examples: Euclidean spaces are not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one. Hamming spaces are bounded due to finiteness. For function spaces with the uniform metric we introduced the notion of uniform boundedness and have proven that this variational property is equivalent to boundedness w.r.t. the particular metric.

Notes on the above can be found here, here, and here.

The whiteboards from past year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.

Lecture 6 (Ac. Year 2023-24)

Sat, 16 Mar 2024 17:35:40 +0300

Notes on the above can be found here, here, here, here.

The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, and here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.

Lecture 5 (Ac. Year 2023-24)

Sat, 09 Mar 2024 01:14:00 +0300

We continued examining subexamples inside the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric.

Notes for the above can be found here and here.

The whiteboards from analogous academic year's (2020-21) lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Lecture 4 (Ac. Year 2023-24)

Sat, 02 Mar 2024 17:33:12 +0300

We have investigated the further relation notion between metrics on the same carrier set; two suchlike metrics are considered equivalent whenever the first dominates the second and vice versa. Hence two such metrics can be considered equivalent-and suspected of introducing "similar properties" on the same carrier-without being necessarily equal as functions.

Notes on the previous can be found here. The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here and here.

Synopsis: Lecture 3 (Ac. Year 2023-24)

Mon, 26 Feb 2024 01:03:28 +0300

Synopsis: Lecture 2 (Ac. Year 2023-24)

Mon, 19 Feb 2024 00:03:24 +0300

After generalizing our basic definitions with the notions of a pseudo-metric and the subsequent notion of a psedo-metric space, as well as with the notion of a metric-subspace, we begun the examination of examples of metrics on spaces comprised of real (finite dimensional) vectors, including the Hamming distance. Our examples showed that several distinct metric spaces (over the same carrier) can become identical when restricted on particular sub-spaces.

Notes on the previous here, and a counterexample of a metric here. The graph-theoretic example can be found here. The whiteboards from the analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here, here and here.

Synopsis: Lecture 1 (Ac. Year 2023-24)

Mon, 12 Feb 2024 02:24:30 +0300

Exercise: Using as carrier set the real line, provide a counterexample of a function that fails to be a metric because it fails to satisfy the triangle inequality, even though it satisfies the remaining properties of the definition.

Synopsis: Lecture 14 (2021-22)

Sun, 05 Jun 2022 00:34:00 +0300

We concluded the course with two applications of the BFPT; those involve the Bellman equation and the Theorem of Picard-Lindelof.

Notes for the above can be found here. The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Synopsis: Lectures 12-13 (2021-22)

Sun, 29 May 2022 18:57:31 +0300

We concluded with further remarks and examples on the notion of Lipschitz continuity. We proceeded with the introduction of some general aspects of fixed point theory, and the examination of the proof and corollaries of the Banach Fixed Point Theorem. We have started our preparations for the examination of applications involving the establishement of the existence and uniqueness of functional equations, by among others proving Blackwell's Lemma.

Synopsis: Lecture 11 (2021-22)

Sun, 22 May 2022 02:21:23 +0300

We have proceeded with the examination of the notions of completeness and Lipschitz continuity.

Notes for the above can be found here and here.

The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Synopsis: Lectures 9th-10th

Sat, 14 May 2022 23:14:30 +0300

We continued with issues of (sequential) convergence in metric spaces. We examined the notion of continuity of functions between metric spaces. We then focused on a major application: the approximation of optimization problems.

Synopsis: Lecture 8 (2021-22)

Sat, 07 May 2022 20:05:08 +0300

We have begun our examination of topological notions in metric spaces with the notion of (sequential) convergence after establishing the Hausdorff and the first countability properties of metric spaces.

Notes on the above can be found here. The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Synopsis: Lecture 7 (2021-22)

Sun, 17 Apr 2022 18:48:49 +0300

We have examined further details on total boundness among others involving, the comparison of metrics, as well as a discriptive examination of the notion of covering numbers. We have pointed out to the usefulness of the above in applications involving issues of convergence in metric spaces, or properties of stochastic properties. We have derived an application of the notion of total boundedness in asymptotic analysis, by deriving a Uniform Law of Large Numbers in a simple framework.

Notes on the above can be found here. Notes on the ULLN application can be found here.

The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Synopsis: Lectures 5-6 (2021-22)

Thu, 07 Apr 2022 14:25:16 +0300

We show that there exists an antimonotone relation between the open (closed) balls of fixed center and radius w.r.t. metrics that obey functional inequalities confirming the aforementioned remark concerning .

We begun our study of metric properties with the finitary notion of boundedness. The balls can be readily used in order to define it as a natural extension of the notion of boundness on the real line (w.r.t. the usual metric). Specifically a subset of a metric space is bounded iff it can be covered by an open (equivalently closed) ball. The center of the covering ball need not be an element of the subset at hand although we have shown that when such a ball exists the center can always be chosen to lie inside this subset.

We have shown that any finite set is (universally) bounded, while the open and closed balls are by construction bounded. The notion is hereditary. If a set is bounded then any subset is also bounded. The dual is evident. If a set is not bounded then any superset is also unbounded. Obviously the notion depends crucially on the metric. For example the discrete space is always bounded. This is in contrast with any Euclidean space (i.e. the equipped with the usual metric) which is not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one, this also implies that the metric space consisting of the set of bounded real functions on X with the uniform metric is not generally bounded (i.e. universally bounded), etc.

We have examined further details on boundness including: several (counter-) examples of boundness in function spaces using uniform boundness, some limiting variance, a characterization of boundedness for metric subspaces, and an equivalent definition that is useful for the introduction of total boundness.

Strenghtening thus, the boundedness characterization we have introduced the notion of total boundness. We have started our examination of the latter. We have given properties, e.g. that the centers of the covering balls can be chosen inside the totally bounded set, as well as examples-counterexamples (the analytical complexity of the notion clearly manifested itself on that counter examples were easier to come up to). Given the analytical complexity of the verification of total boundness, we need the introduction of the notions of covering numbers and metric entropy in order to construct examples and further properties.

Notes on the above can be found here, here, here, here, here, and here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.

Synopsis: Lecture 4 (2021-22)

Sun, 27 Mar 2022 18:49:51 +0300

We continued examining subexamples inside the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric.

We completed our general definitions with the notion of the metric subspace.

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, and obey some monotonicity property. The examples of the real line endowed with the usual metric, the real line endowed with the "exponential metric", the real line endowed with the discrete one, or of the endowed with any of the three "commonly examined" metrics, showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric. We examined the antimonotonic relationship between balls of the same centerd arising by pairs of dominant and dominating metrics.

Notes for the above can be found here and here.

The whiteboards from analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here.

Synopsis: Lecture 3 (2021-22)

Sun, 20 Mar 2022 17:04:06 +0300

(Some of) The examples have shown that it is possible that different metrics on the same carrier set can obey relations, e.g. in the form of functional inequalities. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as with a motivation of further examination of such relations.

We begun the examination of the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and noted that it contains several subexamples, e.g. the spaces of real n-vectors equipped with the max-metric.

Synopsis: Lectures 1-2 (2021-22)

Sun, 13 Mar 2022 17:21:49 +0300

After some brief discussion of the course's scope and aims, and using the overview of the familiar case of the real numbers, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families of metrics, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space. We begun examining of several examples of metrics and subsequent spaces, including the Hamming distance, as well as distances defined on sets of real finite dimensional vectors. We also examined a new example regarding finite simple connected and undirected graphs. We constructed a metric defined on the set of vertices that represents (part of) the graph structure.

Procedural details can be found at the course's syllabus, notes on the previous here, and a counterexample of a metric here. The graph-theoretic example can be found here. The whiteboards from the analogous previous year's lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here and here.

Synopsis 13th Distance Lecture (2020-21)

Sat, 05 Jun 2021 00:59:56 +0300

We concluded the course with two applications of the BFPT; those involve the Bellman equation and the Theorem of Picard-Lindelof.

You can find the lectures' whiteboards here. Notes for the above can be found here.

Synopsis: 12th Distance Lecture

Sat, 29 May 2021 03:23:28 +0300

We further proceeded with the examination of the proof and corollaries for the Banach Fixed Point Theorem. We have started our preparations for the examination of applications by among others proving Blackwell's Lemma.

You can find the lectures' whiteboards here. Notes for the above can be found here.

Synopsis: 11th Distance Lecture

Sun, 23 May 2021 00:12:13 +0300

We concluded with further remarks and examples on the notion of Lipschitz continuity. We then proceeded with the examination of the Banach Fixed Point Theorem after having introduced some general aspects of fixed point theory.

You can find the lectures' whiteboards here. Notes for the above can be found here and here.

Synopsis: 10th Distance Lecture

Sun, 16 May 2021 02:25:16 +0300

We have proceeded with the examination of the notions of completeness and Lipschitz continuity.

You can find the lectures' whiteboards here. Notes for the above can be found here and here.

Synopsis: 9th Distance Lecture

Sun, 25 Apr 2021 17:15:32 +0300

Given our examination of the notion of continuity of functions between metric spaces, we focused on our first major application: approximation of optimization problems.

You can find the lectures' whiteboards here. Notes for the above can be found here.

Synopsis: 8th Distance Lecture

Sun, 18 Apr 2021 01:32:20 +0300

We continued with issues of (sequential) convergence in metric spaces. We examined the notion of continuity of functions between metric spaces. We then focused on our first major application: approximation of optimization problems.

You can find the lectures' whiteboards here. Notes for the above can be found here and here.

Synopsis: 7th Distance Lecture

Mon, 12 Apr 2021 00:51:20 +0300

We have examined further details on total boundness among others involving our discriptive examination of the notion of covering numbers. We have pointed out to the usefulness of the above in applications involving issues of convergence in metric spaces, or properties of stochastic properties. We have begun our examination of topological notions in metric spaces with the notion of (sequential) convergence.

The lecture's whiteboards can be found here. Notes on the above can be found here, and here.

Synopsis: 6th Distance Lecture

Sun, 04 Apr 2021 18:25:35 +0300

Strenghtening the boundedness characterization we have introduced the notion of total boundness. We have moved on to our examination of the latter. Given the analytical complexity of the verification of total boundness, we need the introduction of the notions of covering numbers and metric entropy in order to construct examples and further properties.

The lectures whiteboards can be found here. Notes on the above can be found here, and here.

Synopsis: 5th Distance Lecture

Sun, 28 Mar 2021 13:28:15 +0300

Notes on the above can be found here, here, and here. The lectures whiteboards can be found here.

Synopsis 4th Distance Lecture

Sun, 21 Mar 2021 23:12:36 +0300

We continued our study of ball properties and have shown that the local information that they convey about their center, can be conveyed by a "countable description", as well as that there exists an antimonotone relation between the open (closed) balls of fixed center and radius w.r.t. metrics that obey functional inequalities confirming the aforementioned remark concerning .

We begun our study of metric properties with the finitary notion of boundness. The balls can be readily used in order to define it as a natural extension of the notion of boundness on the real line (w.r.t. the usual metric). Specifically a subset of a metric space is bounded iff it can be covered by an open (equivalently closed) ball. The center of the covering ball need not be an element of the subset at hand although we have shown that when such a ball exists the center can always be chosen to lie inside this subset.

You can find notes for the above here and here.

The distance lecture's whitebords can be found here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.

Synopsis 3rd Distance Lecture

Sun, 14 Mar 2021 23:06:15 +0300

We examined the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and showed that it contains several subexamples, e.g. the spaces of real n-vectors equipped with the max-metric.

We completed our general definitions with the notion of the metric subspace. This as well as the notion of product metric spaces (with a finite number of factors) can be viewed as ways to construct further metric spaces from given one(s), with the resulting metrics carrying relevant information on the given one(s).

We have extended our basic vocabulary regarding balls in metric spaces by showing that those can be used in order to seperate points. This property (which does not generally hold for pseudo-metrics) is essential, since it among others provides with the uniqueness of limits in metric spaces.

Notes for the above can be found here and here.

The lecture's whiteboards can be found here.

Synopsis 2nd Distance Lecture

Sun, 07 Mar 2021 17:22:54 +0300

We went through with the examination of several examples of such spaces, including the Hamming distance, as well as distances defined on sets of real finite dimensional vectors.

Notes on the above can be found here, and the lecture's white boards can be found here.

Synopsis 1st Distance Lecture

Sun, 28 Feb 2021 02:25:43 +0300

After some brief discussion of the course's scope and aims, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space. We begun going through with the examination of several examples of such spaces.

Details can be found at the course's syllabus, notes on the previous here, and a counterexample of a metric here. The lecture's whiteboards can be found here.

Synopsis: Lecture 14-15 (2019-20/Distance Lectures)

Fri, 29 May 2020 03:14:50 +0300

We concluded the course with applications of the BFPT; those involve the Bellman equation and the Theorem of Picard-Lindelof.

You can find the lectures' whiteboards here and here. Notes for the above can be found here.

Synopsis: Lectures 12-13 (2019-20/Distance Lectures)

Fri, 22 May 2020 21:33:17 +0300

We examined a stronger notion of continuity in metric spaces, namely Lipschitz continuity. We proceeded with the examination of the Banach Fixed Point Theorem after having introduced some general aspects of fixed point theory.

You can find the lectures' whiteboards here and here. Notes for the above can be found here and here.

Synopsis: Lectures 10-11 (2019-20/Distance Lectures)

Fri, 15 May 2020 13:59:35 +0300

We examined the notion of continuity of functions between metric spaces. We then focused on our first major application: approximation of optimization problems. We then begun the examination of further non topological notions concerning metric spaces and functions between them: completeness and Lipschitz continuity.

You can find the lectures' whiteboards here and here. Notes for the above can be found here, here and here.

Synopsis: Lecture 9 (2019-20/Distance Lecture)

Sat, 09 May 2020 03:20:12 +0300

We continued with issues of (sequential) convergence in metric spaces.

You can find the lecture's whiteboards here. Notes for the above can be found here.

Synopsis: Lecture 8 (2019-20/Distance Lecture)

Fri, 01 May 2020 21:18:55 +0300

The lecture's whiteboards can be found here. Notes on the above can be found here, and here.

Synopsis: Lectures 6-7 (2019-20/Distance Lecture)

Fri, 10 Apr 2020 05:01:08 +0300

We have examined further details on boundness including some limiting variance and a characterization that is useful for the introduction of total boundness. We have moved on to our examination of the latter. Given the analytical complexity of the verification of total boundness, we have briefly been occupied with the notions of covering numbers and metric entropy.

The lectures whiteboards can be found here and here. Notes on the above can be found here (for a generalization see here), here, here, and here.

Synopsis: Lecture 5 (2019-20/Distance Lecture)

Fri, 03 Apr 2020 03:43:53 +0300

We have examined several examples of boundness in function spaces using uniform boundness, We have also examined the hereditarity of boundness between metrics that satisfy functional inequality relations.

The lectures whiteboards can be found here. Notes on the above can be found here and here (you can also look for a generalization here).

Total Boundness Preparation

Sun, 22 Mar 2020 03:45:21 +0300

Total boundness is a refinement of boundness. The latter is equivalent to the existence of some ε>0 for which there exists a finite cover of open (or equivalently closed) balls for the set at hand. Total boundness strengthens this by requiring that for any ε>0 there exists an analogous finite cover. By definition, for any ε>0, the analogous cover would be essentially determined by the finite set of ball centers, that a. can depend on ε, and b. need not lie inside the set at hand (but we can force them to-how?).

Total boundness is important as it is connected to the determination of a notion of complexity of the space involved, something that is particularly handy when we study spaces comprised of functions. Even though it is not a topological notion, later on we will see that it is related to compactness.

You can find notes for the above here and here.

Exercises corresponding to the first four lectures

Sun, 22 Mar 2020 00:44:08 +0300

You can find here and here exercises corresponding to the notions that were examined in the first three lectures.

Synopsis: Lecture 4 (2019-20)

Sun, 22 Mar 2020 00:42:11 +0300

We have completed our basic vocabulary regarding balls in metric spaces by showing that those can be used in order to seperate points. This property (which does not generally hold for pseudo-metrics) is essential, since it among others provides with the uniqueness of limits in metric spaces.

You can find notes for the above here and here.

Exercise: Show that a subset A of a metric space X is bounded, iff it is a bounded space when considered as metric-subspace of X.

Synopsis: Lecture 3 (2019-20)

Sun, 08 Mar 2020 03:08:52 +0300

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, and obey some monotonicity property. The examples of the real line endowed with the usual metric, the real line endowed with the "exponential metric", the real line endowed with the discrete one, or of the endowed with any of the three "commonly examined" metrics, showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric. We also suspected that the inclusion relations between the balls (with the same center and radius) in w.r.t. the three "commonly examined" metrics may be connected to the functional relations that we have previously derived between the metrics.

You can find notes for the above here and here.

Synopsis: Lectures 1-2 (2019-20)

Mon, 02 Mar 2020 18:46:49 +0300

After some brief discussion of the course's scope and aims, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space. We went through with the examination of several examples of such spaces. We examined the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and hinted that it contains as particular example spaces of real n-vectors equipped with the max-metric.

We have shown that it is possible that different metrics on the same carrier set can obey relations. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as with a motivation of further examination of such relations.

Details can be found at the course's syllabus, notes on the previous here, and a counterexample of a metric here.

Synopsis Lecture 14th

Fri, 31 May 2019 02:48:27 +0300

In the final lecture we initially revisited the issue of the uniform convergence of the self-composition of a contraction inside a complete and bounded metric space in order i) to provide with a more efficient description compared to the one provided in the previous lecture, and ii) to remark that the BFPT essentially implied the pointwise convergence (why?), while the further attribute of boundedness for the space strengthened the latter to uniform convergence. You can find notes for the above here.

We have completed the application involving the existence and uniqueness of a solution to the Bellman equation, and have been further occupied with another application of the Banach FPT, namely the Picard-Lindelof Theorem. You can find notes for the above here.

For purely illustrative purposes we have briefly stated Brouwer's FPT and subsequently examined one application involving the verification of the existence of Nash equilibria in a class of finite non-co-operative games. We have done so in a restricted setting that avoids the use of its generalization to correspendences, i.e. Kakutani's FPT. Our restrictive setting essentially ensures that the best response correspondence is actually a function. You can find notes on the above here. Notice that due to our brief encounter with Brouwer's FPT and the game theoretic application, those two subjects are not included in the exam material. However, Brouwer's FPT can be useful for the solution of Exercise 11 on the Perron-Frobenius Theorem. Should you decide to go for it (something that you are encouraged to do!) it would be useful to go through the material involving the derivation of the particular FPT.

Final TA Session

Thu, 30 May 2019 09:41:50 +0300

In our final TA session we exerted effort into understanding how solutions to Bellman equations relate to solutions of a standard class of Stationary Dynamic Programming Problems. For this purpose, we showed two lemmas related to dynamic programming and used them along with what we have seen in class about bellman equations in continuous and bounded functional space. We saw under what conditions the solution in question exists and is unique. We also solved an example dynamic programming problem. You can find notes on the subject here and in section E4 (pp.233-248) of Ok's textbook (see syllabus).

Finally, we brushed through the matterial covered during the course and cleared things out.

Synopsis Lecture 13th

Sun, 26 May 2019 17:54:03 +0300

We have been occupied with issues involving the uniform convergence of uniformly Lipschitz functions, and the uniform convergence of the self-composition of a contraction inside a bounded and complete metric space (are those two issues somehow related?).

We have begun studying the framework for applications of the BFPT, derived the useful Blackwell's Lemma, and begun the application concerning the uniqueness of the solution to the Bellman equation.

You can find notes for the above here, here (the construction is more effecient compared to the classwork presentation due to the remark c in the notes) and here.

Synopsis Lecture 11th-12th

Sun, 19 May 2019 23:43:32 +0300

We have been initially occupied with the non-topological notion of Lipschitz continuity and the subsequent notion of a contractive self-map.

We have then been occupied with issues on metric fixed point theory involving the derivation of the Banach FPT, some straightforward corrolaries, and some initial applications.

You can find notes on the above here and here.

Third TA Session

Fri, 17 May 2019 16:25:22 +0300

During the third TA session we were mostly preoccupied with proving a lemma concerning the completeness property of general bounded functional spaces (general as in not necessarily mapping on the real numbers), as it is inherited by its image space. You can find the above lemma and its proof here and at the end of these notes here.

We also quickly went through and commented on an economic application of the notions of pointwise versus uniform convergence of sequences of functions and what the distinction can imply for "learning by doing" producrtion processes. You can read more on the topic here.

Synopsis Lectures 9th-10th

Mon, 13 May 2019 14:57:36 +0300

Given the tutorial preparation of the notion of continuity of functions between metric spaces and a brief re-examination of this during the lecture, we have provided with a major application that establishes the continuity of the sup functional when properly restricted to possibly non-empty metric subspaces of the set of bounded real functions equiped with the uniform metric.

Using this and under further assumptions we were occupied with the issue of convergence of approximate maximizers under uniform convergence of their respective criteria.

You can find here notes for the above. We note that the topology generated by the uniform metric might be unecessary strong for the examination of the asymptotic behavior of sequences of maximizers, and a weaker topology for this, is the one of hypo-convergence.

We briefly examined the non-topological notion of completeness of a metric space by first intrducing the notion of a Cauchy sequence inside a metric space. You can find notes for this, here.

Exercise: Try to show that the result on the continuity of the sup operator holds when this is defined over the whole set of bounded functions.

Second TA Session

Fri, 19 Apr 2019 16:45:40 +0300

During the second TA session we were proeccopied with the following:

First, we revisited and corrected exercise 8 of the problem set 1 that was erroneously solved during the first TA session (find the solutions here).

Secondly, we proved two lemmas. The first concerns total boundness and finite product spaces, already seen in class. The proof is analogous to the proof of boundness and finite products. The second concerns the equivalence of characterising continuity using open balls or neigbouring systems (not yet seen in class).

Third, we did not have enough time to solve two exercises. The first is exercise 2 of problem set 3 and it regards the non-total boundness of unit balls on the square integrable functional space. Its solution employes Riesz's Lemma (a short proof of which you may find at O'Searcoid's textbook (see Sylabus) ch. 12), and the Pigeonhole Principle (a principle of Discrete Mathematics). The second, exercise 4 of problem set 3, regards the continuity of metric functions. Its solution is relatively straightforward.

You may find the matterial covered (and not) during the TA session here.

Synopsis 8th Lecture

Fri, 12 Apr 2019 17:12:36 +0300

We were occupied with the comparison of topologies generated by different by different metrics that obey functional inequalities.

We begun the study of the concept of sequential convergence given the notion of a neighborhood system of a point in a topological space and we have proven that in a metric space the notion can be equivalently described via the subsystem of open balls centered at the limit. We have proven that a sequence inside a metric space can have at most one limit, due to the property of separation. The consideration of the behavior of sequences in indiscrete topological spaces implied once more that there exist topologies not generated by metrics. We used the aforemntioned comparison of topologies in order to compare the convergence properies w.r.t. the aforementioned metrics that obey functional inequalities.

Given the notion of sequential convergence we begun to examine of the notion of continuity of functions between metric spaces.

You can find notes for the above here and here.

Synopsis 7th Lecture

Sat, 06 Apr 2019 22:28:03 +0300

We concluded our examination of the notion of total boundness by considering issues involving metric comparisons and finite products. We provided several remarks regarding hereditarity, etc, that implied that the notion coincides with that of boundness on Euclidean spaces.

We begun the examination of topological notions on a metric space that arise by the presence of a metric. We initially defined the notion of open and closed subsets as duals, then the subsequent notion of a topology arising from a metric, and then generalized the notion in order to see that there exist topologies that do not arise from metrics (e.g. the indisrete topology when the carrier set has more than one elements). We have that when a topology arises from a metric is termed metrizable (e.g. the discrete topology that arises from the discrete metric). The issue of metrization of a topology is essentially settled by the Nagata-Smirnov theorem that is completely outside the scope of the course.

You can find notes for the above here and here.

First TA session

Sat, 30 Mar 2019 19:54:14 +0300

On our first TA session we went through two sets of exercises on some elementary notions of metric spaces, also covered during the lectures. In particular we tried to firmly embed and understand the properties of metric functions and open (closed) balls in metric spaces.

You can find here and here solutions to all the exercises that we saw during the TA session. As mentioned in class, the rest of the exercises are either similar to those we solved or will not preoccupy as in general.

As was spotted during the TA session, the solution to exercise 8 of the first problem set was erroneous. Additionally, the discussion on exercise 12 fell short of a proper answer. In the solutions linked above you can find better answers to both. We will rectify those shortcomings in the next TA session.

Synopsis 5th-6th Lecture (2018-19)

Sun, 24 Mar 2019 18:30:59 +0300

We have been occupied with further examples, and the issues of equivalence of metrics w.r.t. boundness, and the "invariance" of boundness w.r.t. finite products and the metrics we have defined on such products. You can find notes for the above here and here.

We have also been occupied with an example of a "sequence of metric spaces", where the property is "somehow lost in the limit", that can be found here. You can also find here a generalization of the issue of metrics comparison w.r.t. boundness, and here a result on boundness obtained by some asymptotic comparison of balls between two metrics.

We moved on to a refinement of the notion, obtaining the notion of total boundness. If we can perceive boundness as equivalent to the existence of some ε>0 for which there exists a finite cover of open (or equivalently closed) balls for the set at hand, total boundness strengthens this by requiring that for any ε>0 there exists an analogous finite cover. By definition, for any ε>0, the analogous cover would be essentially determined by the finite set of ball centers, that a. can depend on ε, and b. need not lie inside the set at hand. We begun working on the notion by establishing that the version of the definition involving covers consisting of open balls is equivalent to the version of the definition involving covers of closed balls.

We continued establishing first that total boundness implies boundness, and then moved on to examine issues of hereditarity, of the universal total boundness for finite sets, of the definition of a totally bounded space, the subsequent construction of counter-examples, and then showed that the notion is generally stronger (as expected) than the usual boundness, by showing that a discrete space is totally bounded iff it is finite, and by providing a(n) (applicationwise more interesting) example of a subset of a function space that is uniformly bounded but not totaly bounded w.r.t. the uniform metric.

You can find notes for the above here.

Synopsis: 4th Lecture (2018-19)

Mon, 18 Mar 2019 20:00:59 +0300

We begun our study of metric space properties with the finitary notion of boundness. The existence of the collection of the open balls in a metric space allows for the definition of the concept of a bounded subset (via an obvious extension of the analogous definition in the real line w.r.t. the usual metric). It holds iff it can be covered by an open (equivalently closed) ball, while the center of the covering ball need not be an element of the subset at hand. We have shown that any finite set is (universally) bounded, while the open and closed balls are by construction bounded. The notion is hereditary. If a set is bounded then any subset is also bounded. The dual is evident. If a set is not bounded then any superset is also unbounded. Obviously the notion depends crucially on the metric. For example the discrete space is always bounded. This is in contrast with any Euclidean space (i.e. the equipped with the usual metric) which is not bounded-the same is true when equipped with any of the metrics we have been discussing except for the discrete one, this also implies that the metric space consisting of the set of bounded real functions on X with the uniform metric is not generally bounded, etc.

You can find notes for the above here and here.

Exercises Corresponding to the Notions of Lectures 1-3

Sun, 10 Mar 2019 23:33:00 +0300

You can find here and here exercises corresponding to the notions that were examined in the first three lectures.

Synopsis: 3rd Lecture (2018-19)

Sun, 10 Mar 2019 23:29:02 +0300

We completed our general definitions with the notions of the metric subspaces and of the product metric spaces (with a finite number of factors). We can view the above as ways to construct further metric spaces from given one(s), with the resulting metrics carrying relevant information on the given one(s).

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, obey some monotonicity property, and they can separate points in a metric space, while this does not generally hold for pseudo metrics. The examples of the real line endowed with the usual metric, the real line endowed with the "exponential metric", the real line endowed with the discrete one, or of the endowed with any of the three "commonly examined" metrics, showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric. We also suspected that the inclusion relations between the balls (with the same center and radius) in w.r.t. the three "commonly examined" metrics may be connected to the functional relations that we have previously derived between the metrics.

You can find notes for the above here and here.

Synopsis: 2nd Lecture (2018-19)

Fri, 01 Mar 2019 19:35:14 +0300

We examined the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and showed that it contains as particular example the set of bounded real sequences with the relevant extension of the finite dimensional max-metric. We have also examined the subset of the latter consisting of the square summable real sequences and noticed that we can also define in this the obvious extension of the Euclidean metric. Furthermore, we have also examined the subset of the latter consisting of the absolutely summable real sequences and noticed that we can also define in this the obvious extension of the absolute metric. Notice that the last three examples essentially showed us that even though each of the Euclidean, max and absolute metric are essentially extensions of the absolute valued metric on the reals to "finite dimensional" Euclidean spaces, we must be careful when we try to further extend them to "infinite dimensional spaces".

You can find notes for the above here.

Synopsis: 1st Lecture (2018-19)

Sat, 23 Feb 2019 02:32:03 +0300

After some brief discussion of the course's scope and aims, as well as of some prerequisite notions, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space. We have begun the examination of several examples of such spaces. Details can be found at the course's syllabus, notes on the previous here, and a counterexample of a metric here.

Synopsis: Lecture 13th (2017)

Sat, 27 May 2017 16:50:16 +0300

We have been further occupied with Brouwer's FPT providing an example that showed that it cannot generally hold in infinite dimensional spaces. We have subsequently examined an application of Brouwer's FPT application for the verification of the existence of Nash equilibria in a class of finite non-co-operative games, in a restricted setting that avoids the use of its generalization to correspendences, i.e. Kakutani's FPT. Our restrictive setting essentially ensures that the best response correspondence is actually a function. You can find notes on the above here.

We have also examined an application of Dudley's Theorem in the verification of the continuity of almost all of the sample paths of a standard Wiener process. You can find notes of this here (as noted in the lecture the particular application of Dudley's Theorem will not be a part of your exam material).

Synopsis: Lecture 12th (2017)

Sat, 20 May 2017 00:18:36 +0300

We have been further occupied with the development of preparatory notions for the Brouwer's FPT. Those involved the topological notion of retraction. We have provided examples and counter examples. We have been examining the No Retraction Theorem-Borsuk's Lemma. We provided with a sketch of the proof using the concept of singular homology. Obviously this sketch of proof is out of the scope of the course. Using the latter theorem and the other preparatory notions we have proven the Brouwer Fixed Point Theorem. We have briefly examined an application in linear algebra, namely the Perron-Frobenius Theorem (the proof of which is an optional exercise). You can find notes for them here and here. (Notice that the construction of a retraction from a set in a Euclidean space to a non empty compact and convex subset of it involved the convexity of the Euclidean norm. A further detail is also used that essentially implies that if the objective function in the definition of the retraction has more than one minimizers then those would constitute a convex set that would essentially lie on the boundary of an Euclidean closed balls. But Euclidean spheres have only trivial convex subsets).

Synopsis: Lecture 11th (2017)

Sat, 13 May 2017 20:27:23 +0300

We have been occupied with a further application of the Banach FPT, namely the Picard-Lindelof Theorem. (Further corrected and supplemented) Notes of which you can find here. We begun our preparation for the establishment of the Brouwer FPT, introducing the notions of fixed point property and of that of topological homeomorphism. You can find (corrected) notes for the above here.

Synopsis: Lectures 9th-10th (2017)

Sat, 06 May 2017 18:33:55 +0300

We have been occupied with issues about metric fixed point theory involving the derivation of the Banach FPT, and applications w.r.t. the properties of the Bellman equation. You can find (corrected and supplemented) notes on the above here.

Synopsis: 8th Lecture (2017)

Fri, 28 Apr 2017 14:38:04 +0300

Given the examination of the non-topological notion of completeness of a metric space that was made available in the previous tutorial we were occupied with the issue of completeness (w.r.t. the relevant uniform metric) of a bounded function space when the range is a complete metric space. You can find notes for the aforementioned notions here (remember that the part of the proof of the latter result concerning the boundness of the limit of the relevant Cauchy sequence was left as an exercise).

We have also studied the stronger than topological continuty notion of Lipschitz continuity, obtaining finally the notion of a contraction. You can find for those notions here. We are now almost ready to study our first fixed point theorem.

Synopsis: 7th Lecture (2017)

Sat, 08 Apr 2017 18:49:49 +0300

Using this and under further assumptions we were occupied with the issue of convergence of approximate maximizers under uniform convergence of their respective criteria.

Synopsis: 6th Lecture (2017)

Fri, 31 Mar 2017 15:21:13 +0300

Given the general definition of a topological space we defined the concept of a metrizable topology (i.e. a topology generated by a metric). We derived an easy example concerning the indiscrete topology that showed the existence of topologies that are not metrizable. We were occupied with the issue of when two metrics define the same topology using the usual concepts of comparisons between metrics.

We begun the study of the concept of sequential convergence given the notion of a neighborhood system of a point in a topological space and we have proven that in a metric space the notion can be equivalently described via the subsystem of open balls centered at the limit. We have proven that a sequence inside a metric space can have at most one limit, due to the property of separation. The consideration of the behavior of sequences in indiscrete topological spaces implied that there exist topologies not generated by metrics. Hence we have proven that closeness of a subset is equivalent to that every convergent sequence with elements in the subsets converges inside the subset. While closeness implies this sequential property in every topological space, the converse is true in metric spaces due to first countability. We were occupied with the issue of when two different metrics imply the same asymptotic behavior for the same sequence using again the usual notions of comparison.

You can find notes for the above here.

Addendum-5th Lecture (2017)

Sat, 25 Mar 2017 04:10:28 +0300

You can find here an example of the use of the metric entropy integral in the case of the Wiener process. We will work out this example when we examine the notion of continuity of functions between metric spaces. Furthermore, you can find here, a simple example, built upon an already examined one, that makes evident that non total boundness does not imply the existence of one element balls (as is typically the case in discrete spaces). Furthermore it implies that open balls can be non totally bounded, even though they are by construction bounded.

Synopsis: 5th Lecture (2017)

Fri, 24 Mar 2017 20:30:39 +0300

We continued the examination of the notion of total boundness. We provided several remarks regarding hereditarity, etc, that implied that the notion coincides with that of boundness on Euclidean spaces. We briefly discussed the notion of metric entropy and described the Metric Entropy Integral Theorem of Dudley as an astonishing application of the notion of total boundness in the theory of Gaussian Processes. As previously you can find notes for total boundness here and for the aforementioned remarks and discussions see also here (notice that the latter contains the correction of the system of inequalities for the covering numbers w.r.t. the particular equivalence between different metrics compared to what was derived in class).

Exercise: Prove that the metric appearing in Dudley's Theorem is actually a pseudo-metric (and notice that the notions of boundness and total boundness can be readily extended to pseudo-metrics).

We begun the examination of topological notions on a metric space that arise by the presence of a metric. We initially defined the notion of a topology on a non-empty set and subsequently the notion of open and closed subsets as duals. You can find notes for the above here.

Boundness: Further Remarks and Asymptotic Comparisons

Sat, 18 Mar 2017 01:07:13 +0300

You can find here a generalization of the issue of metrics comparison w.r.t. boundness, here an example where the property is "somehow lost in the limit", and here a result on boundness obtained by some asymptotic comparison of balls between two metrics.

Synopsis: 4th Lecture (2017)

Fri, 17 Mar 2017 16:36:46 +0300

We have been occupied with further examples, and the issues of hereditarity on subsets of boundness, equivalence of metrics w.r.t. boundness, and the "invariance" of boundness w.r.t. finite products and the metrics we have defined on such products. You can find notes for the above here, here, and here.

We moved on to a refinement of the notion, obtaining the notion of total boundness, and we have initially provided with the non-equivalence of the two notions. You can find notes for the above here.

Balls Non-convexities

Sun, 12 Mar 2017 02:22:23 +0300

You can find here the example of a metric space with a metric defined in such a way so that some of the open balls are non-convex (the carrier set is simply the reals-hence they have the algebraic structure for the notion of convexity to make sense), along with a generalization. This has emerged by a discussion with one of your collegues.

Synopsis: 3rd Lecture (2017)

Fri, 10 Mar 2017 02:32:38 +0300

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, obey some monotonicity property, they can separate points in a metric space, while this does not generally hold for pseudo metrics, and the local information that they convey about their center, can be conveyed by a "countable description". The example of the real line endowed with the usual metric and the real line endowed with the discrete one showed that the “form” of the open (and/or the closed) balls crucially depends on the metric.

A finitary property that the existence of balls in a metric space make examinable, is the notion of boundness. The existence of the collection of the open balls in a metric space allows for the definition of the concept of a bounded subset. The open and closed balls are essentially considered bounded and the notion is also considered as hereditary. The center of the covering ball need not be an element of the subset at hand. Obviously the notion depends crucially on the metric. For example the discrete space is always bounded. This is in contrast with any Euclidean space (i.e. the equipped with the usual metric) which is not bounded (the same is true when equipped with any of the metrics we have been discussing except for the discrete one-this also implies that the metric space consisting of the set of bounded real functions on X with the uniform metric is not generally bounded.). Obviously the notion is hereditary. If a set is bounded then any subset is also bounded. The dual is evident. If a set is not bounded then any superset is also unbounded. Any finite subset of a metric space is bounded, etc.

Synopsis: 2nd Lecture (2017)

Sat, 04 Mar 2017 06:04:42 +0300

We examined the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and showed that it contains as particular examples with the max-metric, or the set of bounded real sequences with the analogous metric. We have also examined the subset of the latter consisting of the squared summable real sequences and noticed that we can also define in this the obvious extension of the Euclidean metric. Furthermore, ee have also examined the subset of the latter consisting of the absolutely summable real sequences and noticed that we can also define in this the obvious extension of the absolute metric.

We were also occupied with the issues of metric subspaces and of product metric spaces (with a finite number of factors).

Synopsis: 1st Lecture (2017)

Fri, 24 Feb 2017 05:33:15 +0300

After some brief discussion of the course's scope and aims, as well as of some prerequisite notions, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space. We have begun the examination of several examples of such spaces. Details can be found at the course's syllabus, notes on the previous here, and a counterexample of a metric here.

Synopsis: 13th Lecture

Fri, 27 May 2016 01:14:01 +0300

We have been occupied with Brouwer's FPT and its subsequent application for the verification of the existence of Nash equilibria in a class of finite non-co-operative games, in a restricted setting that avoids the use of its generalization to correspendences, i.e. Kakutani's FPT. You can find notes on the above here.

Synopsis: 12th Lecture

Sun, 22 May 2016 12:56:24 +0300

We have been occupied with the development of preparatory notions for the Brouwer's FPT. You can find notes for them here. (Notice that the sketch of proof of Borsuk's Lemma is out of the scope of the lectures. Anyhow, the notes contain a correction of the sketch as presented in the class. Specifically, the isomorphism between the homology groups of the same order does not hold for retracts in general, but refers to the stronger notion of deformation retracts. Check the notes for the corrected argument).

Synopsis: 10th-11th Lectures

Mon, 16 May 2016 01:43:44 +0300

We have been occupied with issues involving the Banach FPT, generalizations and applications involving the properties of the Bellman equation and Picard's Theorem. Yoy can find notes on the above here.

Synopsis: 9th Lecture

Fri, 22 Apr 2016 01:12:05 +0300

We have continued the examination of the non-topological notion of completeness of a metric space. You can find here notes for it. We have also studied the stronger than topological continuty notion of Lipschitz continuity, obtaining finally the notion of a contraction. You can find here notes for it.

Synopsis: 7th-8th Lectures

Fri, 15 Apr 2016 04:33:52 +0300

We continued the examination of issues involving convergence and continuity. After a little bit of trouble, we have provided with an application that establishes the continuity of the sup functional when properly restricted to possibly non-empty metric subspaces of the set of bounded real functions equiped with the uniform metric. Using this and under further assumptions we were occupied with the issue of convergence of approximate maximizers under uniform convergence of their respective criteria. You can find here notes for the above. We note that the topology generated by the uniform metric might be unecessary strong for the examination of the asymptotic behavior of sequences of maximizers, and a weaker topology for this, is the one of hypo-convergence.

We have begun the examination of the non-topological notion of completeness of a metric space. You can find here notes for it.

Synopsis: 6th Lecture

Fri, 01 Apr 2016 04:12:42 +0300

We continued the study of the concept of sequential convergence. We have proven that it can be equivalently described by systems of open neighborhoods, hence it is a notion present in more general topological spaces. We have proven that a sequence inside a metric space can have at most one limit, due to the property of separation. The consideration of the behavior of sequences in indiscrete topological spaces implied that there exist topologies not generated by metrics. We discussed the issue of characterization of topological properties via sequential convergence. Hence we have proven that closeness of a subset is equivalent to that every convergent sequence with elements in the subsets converges inside the subset. While closeness implies this sequential property in every topological space, the converse is true in metric spaces due to first countability. We were occupied with the issue of when two different metrics imply the same asymptotic behavior for the same sequence.

We begun the examination of continuity of functions "between" metric spaces. We provided of a sequential characterization of continuity at a point, as a property that does not destroy convergence to the point. We also provided equivalent characterizations via open balls and systems of open neighborhoods. The latter characterization implies that this notion is more generic, in that it is definable for functions between general topological spaces.

Synopsis: 5th Lecture

Fri, 01 Apr 2016 04:03:16 +0300

We continued with the examination of the notion of total boundness. We proved that it is a refinement of the notion of boundness, in the sense that total boundeness implies boundness, while we refered to their equivalence in Euclidean spaces. We were occupied with the issues of the equivalence of metrics w.r.t. total boundness, and the invariance of the notion w.r.t. finite products.

We begun the examination of topological notions on a metric space that arise by the presence of a metric. We defined open and closed subsets as duals, and considered the collection of open subsets as a topology generated by a metric. We were occupied with the issue of when two metrics define the same topology. We begun the study of the concept of sequential convergence.

Synopsis: 4th Lecture

Thu, 17 Mar 2016 17:51:15 +0300

We have been occupied with issues of hereditarity on subsets of boundness, equivalence of metrics w.r.t. boundness, and the "invariance" of boundness w.r.t. finite products and the metrics we have defined on such products. We moved on to a refinement of the notion, obtaining the notion of total boundness, and we have initially provided with the non-equivalence of the two notions.

Synopsis: 3rd Lecture

Mon, 14 Mar 2016 20:49:20 +0300

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, obey some monotonicity property, they can separate points in a metric space, and the local information that they convey about their center, can be conveyed by a "countable description". The example of the real line endowed with the usual metric and the real line endowed with the discrete one showed that the “form” of the open (and/or the closed) balls crucially depends on the metric.

Synopsis: 2nd Lecture

Fri, 04 Mar 2016 00:17:38 +0300

We have been occupied with further examples of metrics and subsequent metric spaces, and have shown that it is possible that different metrics on the same carrier set can obey relations. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as with a motivation of further examination of such relations. We examined the important example of the space of bounded real functions on a non-empty domain endowed with the uniform metric and showed that it contains as particular examples with the max-metric, or the set of bounded real sequence with the analogous metric. We were also occupied with the issues of metric subspaces and of product metric spaces (with a finite number of factors).

Synopsis: 1st Lecture

Fri, 26 Feb 2016 18:43:13 +0300

After some brief discussion of prerequisite notions, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, some of which attribute possibly different properties to the reference (or carrier) set. The structured set comprised of the carrier set with the metric was defined as a metric space. We have begun the examination of several examples of such spaces.