## Functional analysis front page

### Quick description

Functional analysis is the study of infinite-dimensional vector spaces. Typically these are spaces of functions such as being the space of functions where for . These spaces must have a topology, which is often given in terms of a norm such as The name comes from the notion of functionals which are continuous linear functions where is the space concerned. A major early problem was the identification of ways of representing the functionals of a given infinite dimensional space . For example, the Riesz representation theorem (or perhaps one should say theorems) says that if is the space of continuous functions , then any functional , with the norm on , could be represented in terms of a function of bounded variation with where the integral is understood in the Riemann-Stieltjes sense. Related results concern, for example, the functionals on spaces.

Such analysis quickly led to the classification of infinite-dimensional spaces: Banach spaces, Hilbert spaces, reflexive spaces, and investigation of their properties.

This is a large area and involves large parts of classical and modern analysis. It is used in a large number of subjects, but especially partial differential equations, geometric analysis, quantum mechanics, and dynamical systems, to give a small sample of relatively applied parts of mathematics that use functional analysis.

### Prerequisites

Calculus, basic analysis.

### General discussion

Here is a (small) sample of topics:

• representation of functionals

• dual spaces, topologies

• different notions of convergence: strong, weak, weak*

• topological vector spaces

• operator algebras

• spaces of (linear) operators • geometry of Banach spaces

• spaces of measures with values in (Banach) spaces (see vector measure)

• existence of solutions to operator equations: solve for • fixed point and minimax theorems

• Fredholm operators and Fredholm index

• compactness of sets in specific spaces

• optimization of functions defined on spaces (see Convex and variational analysis)

• differential equations on spaces and evolution equations

• convex and variational analysis

• interpolation spaces

• particular (classes of) spaces: e.g., Sobolev spaces, Orlicz spaces, , , 