FUNCTIONAL ANALYSIS

The course contributes to acquiring the theoretical and logical skills necessary for the education of a Mathematics graduate. In particular, it provides the foundational tools of Functional Analysis, useful for those who wish to pursue research. The course will present broader structures compared to those the student already knows from previous studies, such as topological vector spaces and locally convex spaces. A thorough study of Banach spaces and operators between Banach spaces will be undertaken. Weak topologies and particularly important classes of Banach spaces, such as reflexive spaces, will also be covered.

Specifically, according to the Dublin descriptors, the objectives are as follows:

Knowledge and Understanding: The student will become familiar with the fundamental concepts and classical theorems of Functional Analysis and some important classes of spaces, such as reflexive spaces. They will learn to operate within topological vector spaces and with linear and continuous operators between them, and to use weak topologies.

Applying Knowledge and Understanding: The student will be able to apply the general results learned to solve some theoretical and/or technical exercises.

Making Judgements: The student will be encouraged to independently study some results not covered during the lectures and to present them in a seminar.

Communication Skills: The student will learn to present the course content clearly, precisely, and concisely, with rigor and critical insight.

Learning Skills: The student will be able to reflect on proofs and master techniques that can be useful for tackling other problems.

Topological Vector Spaces.  Definition and characterization of vector topologies. Characterization of Hausdorff topological vector spaces. Locally convex topological vector spaces and their characterization. Topology of uniform convergence on compact subsets of $C(S)$, with $S$ open in ${\mathbb{R}}^n$. Non-locally convex vector topology on $C([0,1])$. Metrizability of locally convex spaces. Normability of a topological vector space. Normed spaces. Banach spaces. Hausdorff topological vector spaces with finite dimension. Riesz's characterization of finite-dimensionality in a normed space. Minkowski functional. Hahn-Banach theorem and its corollaries. Separation theorems.

Linear Operators and Functionals.  Various criteria for continuity of linear operators and functionals. The space of linear and continuous operators between two normed spaces. The Open Mapping Theorem and applications. The Closed Graph Theorem. The Uniform Boundedness Principle. Banach-Steinhaus theorem. Adjoint of an operator.

Weak Topologies. The weak topology of a Hausdorff locally convex topological vector space. Coincidence of the closure and weak closure of a convex set. Mazur's theorem. Minimization of lower semicontinuous quasi-convex functionals on weakly compact sets. Comparison between the strong topology, weak topology, and weak-* topology in the dual space of a normed space. Krein-Smulyan theorem. Eberlein-Smulyan theorem. Characterization of finite-dimensionality of a normed space through the coincidence of strong and weak topologies. Banach-Alaoglu theorem. Goldstein theorem. 

Schauder basis. Definition of Schauder basis. Basic sequences. Mazur's techniques for constructing basic sequences. Shrinking Schauder bases and boundedly complete bases. Weakly unconditional Cauchy series. Pelczynski's $c_0$-theorem. Ramsey's theorem. Rosenthal's $l_1$- theorem.

Reflexive Banach spaces. Kakutani and James's characterizations of reflexive Banach spaces. Characterization of reflexive and separable Banach spaces. Metrizability of weakly compact sets in separable normed spaces. Separability and weak topologies. Uniformly convex spaces. Milman-Pettis theorem.

Geometric aspects of the Radon-Nikodym property. Convex sets with the Radon-Nikodym property. Extreme points and the Krein-Milman property. Exposed points, strongly exposed points, and support points. Density of support functionals. Lindenstrauss's theorem. Sets that lose the Krein-Milman property. Dentable points. Theorem of Huff-Morris-Davis-Phelps.

1. R.D.  Bourgin, Geometrical aspects of convex sets with the Radon-Nikodym property. LNM 993 Springer-Verlag. (1983)

2. J. Diestel, Geometry of Banach spaces - selected topics. LNM 485  Springer-Verlag. (1975)

3. J. Diestel, Sequences and series in Banach spaces. Springer-Verlag. (1984)

4. J. Horvath, Topological vector spaces and distributions. Addison-Wesley. (1966)

5. R. E. Megginson, An introduction to Banach space theory. Springer-Verlag. (1998)

Assessment Methods:

The exam consists of an oral test. The following criteria will generally be used to assign grades:

• Fail (below 18): The student has not acquired the basic concepts.
• 18-23: The student has acquired the basic concepts. Their ability to apply the knowledge gained and to present the content with logical rigor and critical insight is barely sufficient.
• 24-27: The student has a good grasp of the course content, presenting it with a good level of logical rigor and critical insight. They have good abilities to connect the learned content.
• 28-30 with honors: The student has mastered all the course content and can reflect on the proofs and master techniques useful for tackling other problems. They have excellent communication and learning skills and a remarkable ability to connect the learned content.

Students with disabilities and/or Specific Learning Disabilities (SLD) are encouraged to discuss any necessary accommodations with the instructor based on their specific needs. They can also contact the CInAP (Center for Support for Students with Disabilities) representative of the Department of Mathematics.

The questions listed below do not constitute a comprehensive list but are only a few examples.

1. Geometric version of the Hahn-Banach theorem and separation theorems
2. Reflexive spaces and their characterization
3. Example of a topological space that is not locally convex
4. Example of a convex set without the Krein-Milman property
5. James's theorem