# FUNCTIONAL ANALYSIS

**Academic Year 2016/2017**- 2° Year - Curriculum A

## Learning Objectives

The main aim of the course is to provide the student with a thorough treatment of the general structures on which the most advanced developments of Analysis are based.

In particular, it will be emphasized how certain topics known to the student find in those structures their most natural and definitive setting.

Also, it will be stimulated the student ability of applying the knowledgement and understanding of the constructed theory to specific problems.

## Detailed Course Content

Topological vector spaces. The Minkowski functional. Locally convex topological vector spaces. Fréchet spaces. Normed spaces.

The Kolmogoroff criterion on normability. The Hahn-Banach theorem. Separation theorems. Extreme points of a convex set. The Krein-Milman theorem.

Linear operators. The space of all continuous linear operators between two normed spaces. The open mapping theorem. The continuous inverse theorem.

The two-norms theorem. The closed graph theorem. The uniform boundedness principle. The Banach-Steinhaus theorem.

Weak topologies. Coincidence of the strong closure and the weak closure of a convex set. Strong, weak and weak* topologies on the dual of a normed space.

The Krein-Smulyan theorem. The Eberlein-Smulyan theorem.

Polars. The bipolar theorem. The Banach-Alaoglu theorem. The canonical mapping between a normed space and its bidual. The Goldstine theorem.

Reflexive Banach spaces. The Kakutani theorem. The James theorem. Characterization of reflexive and separable Banach spaces. Metrizability of

weakly compact sets in separable normed spaces. Uniformly convex normed spaces. The Milman-Pettis theorem. Minimization theorems in reflexive

Banach spaces.

Inner product spaces. The Cauchy-Schwarz inequality. Hilbert spaces. Representation of continuous linear functionals on a Hilbert space. Orthonormal

sets. The Bessel inequality. The Parseval identity. The Riesz-Fischer theorem. Existence of orthonormal bases in separable Hilbert spaces.

Basic theorems on compact linear operators in Banach spaces.

The Brézis-Browder principle. The Ekeland variational principle and some of its applications.

## Textbook Information

1. L. V. Kantorovich, G. P. Akilov, Functional analysis, Pergamon Press.