FUNCTIONAL ANALYSIS

The course contributes to the acquisition of the theoretical and logical competencies required for the education of a Mathematics graduate. In particular, it provides the fundamental tools of Functional Analysis, which are essential for students intending to pursue research activities. More general structures than those encountered in previous studies will be introduced, including topological vector spaces and locally convex spaces. A thorough study of Banach spaces and operators between Banach spaces will be conducted. Weak topologies and particularly important classes of Banach spaces, such as reflexive spaces, separable spaces, and spaces with bases, will also be presented.

More specifically, according to the Dublin Descriptors, the learning objectives are as follows:

Knowledge and Understanding

Students will acquire knowledge of the fundamental concepts and classical theorems of Functional Analysis, as well as some important classes of spaces, such as reflexive spaces. They will learn to work with topological vector spaces and with linear continuous operators between them, and to use weak topologies.

Applying Knowledge and Understanding

Students will be able to apply the general results learned during the course to the solution of theoretical and/or technical exercises.

Making Judgements

Students will be encouraged to independently study selected results not covered during lectures and to present them in a seminar.

Communication Skills

Students will learn to present the course contents clearly, accurately, and concisely, with rigor and a critical approach.

Learning Skills

Students will be able to critically reflect on mathematical proofs and master techniques that may be useful in tackling other mathematical problems.

Attendance of lectures is strongly recommended (see the Degree Programme Regulations).

Topological Vector Spaces. Definition and characterization of vector topologies. Characterization of Hausdorff topological vector spaces. Locally convex topological vector spaces and their characterization. The topology of uniform convergence on compact subsets of 

C(S), where S is an open subset of R^n. A vector topology on C([0,1]) that is not locally convex. Metrizability of locally convex spaces. Normability of a topological vector space. Normed spaces. Banach spaces. Finite-dimensional Hausdorff topological vector spaces. Riesz’s characterization of finite-dimensional normed spaces. Minkowski functional. Hahn-Banach Theorem and its corollaries. Separation theorems.

Linear Operators and Functionals. Various continuity criteria for linear operators and functionals. The space of continuous linear operators between two normed spaces. The Open Mapping Theorem and its applications. The Closed Graph Theorem. The Uniform Boundedness Principle. The Banach-Steinhaus Theorem. Adjoint operators. Kernel and range of an operator. Compact, weakly compact, and completely continuous operators. Gantmacher’s Theorem. Schauder’s Theorem. The Davis-Figiel-Johnson-Pełczyński Theorem. Continuous linear projections and complemented subspaces. An example of a non-complemented subspace.

Weak Topologies. The weak topology of a Hausdorff locally convex topological vector space. Coincidence of the norm closure and weak closure of a convex set. Mazur’s Theorem. Comparison among the strong topology, the weak topology, and the weak-* topology on the topological dual of a normed space. Krein-Šmulian Theorem. Day’s Theorem. Eberlein-Šmulian Theorem. Characterization of finite-dimensional normed spaces through the coincidence of the strong and weak topologies. Banach-Alaoglu Theorem. Goldstine’s Theorem.

Schauder Bases. Definition of Schauder bases. Basic sequences. Mazur’s techniques for constructing basic sequences. Shrinking and boundedly complete Schauder bases. Weakly unconditionally Cauchy series. Pełczyński’s c_0 -Theorem. Ramsey’s Theorem. Rosenthal’s ℓ1 -Theorem.

Reflexive Banach Spaces. Bishop-Phelps Theorem. James’ Theorem on weak compactness. James’ characterizations of reflexive Banach spaces. Characterization of reflexive Banach spaces with Schauder bases. Metrizability of weakly compact sets in separable normed spaces. Separability and weak topologies. Uniformly convex spaces. Milman-Pettis Theorem.

Geometric Aspects of the Radon-Nikodým Property. Convex sets with the Radon-Nikodým Property. Extreme points and the Krein-Milman Property. Exposed points, strongly exposed points, and support points. Density of support functionals. Lindenstrauss’ Theorem. Sets failing the Krein-Milman Property. Dentable sets. The Huff-Morris-Davis-Phelps Theorem.

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)

6.  H.H. Schaefer, Topological Vector spaces, Springer

The questions listed below aren't exhaustive, but  just a few examples .

1. Hahn Banach Theorem:  geometric form and applications.

2. Reflexive Banach spaces and some of their characterizations.

3. Example of a topological space that is not locally convex space