FUNCTIONAL ANALYSIS

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.

In more detail, following the Dublin descriptors, the objectives are the following:

Knowledge and understanding: the student will learn to operate in topological vector spaces and with continuous linear operators between them.

Applying knowledge and understanding: the student will be guided in the ability to realize applications of the general results gradually established.

Making judgements: the student will be stimulated to study autonomously some results not developed during lessons.

Communication skills: the student will learn to expose in a clear, rigorous and concise manner.

Learning skills: the student will be able to face exercices and found proofs of simple results.

The course will be performed through frontal lessons. If necessary, the telematic way will be adopted. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus. Learning assessment may also be carried out online, should the conditions require it.

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.

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