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Introduction to Functional Analysis

The Functional analysis is a category of mathematical analysis, that is formed by the study of vector spaces gifted with some kind of limit-related structure examples are inner product, norm, topology etc. and the linear operators that are acting upon these spaces and respecting these structures in an appropriate sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators among the function spaces. This view point turned out to be particularly helpful for the study of differential and integral equations.

The usage of the word functional goes back to the calculus of variations that implies a function whose argument is a function and the name was first used in Hadamard's in the year 1910. The common concept of functional had formerly been introduced in the year 1887 by the Italian mathematician and physicist Vito Volterra. The theory of non-linear functional was sustained by the students of Hadamard, mainly Levy and Frechet. The Hadamard also founded the modern school of linear functional analysis which is further developed by Riesz and by the group of Polish mathematicians around Stefan Banach.

In modern introductory the functional analysis is the subject which is seen as the study of vector spaces endowed with a topology, specific in infinite dimensional spaces. In difference with the linear algebra which deals mostly with finite dimensional spaces or do not use topology. A necessary part of functional analysis is the extension of the probability, theory of measure, and integration to infinite dimensional spaces; this is also termed as infinite dimensional analysis.

**Normed vector spaces**

The basic first class spaces which are studied in functional analysis are complete Normed vector spaces over the real or complex numbers. That kind of spaces is known as Banach spaces. A significant instance is a Hilbert space, where the norms approach from an inner product. Such spaces have fundamental significance in several regions, including the mathematical formulation of quantum mechanics.

More typically, functional analysis involves the study of Frechet spaces and other topological vector spaces not endowed with a norm.

A significant object of study in functional analysis is the continuous linear operators which are defined on Banach and Hilbert spaces. This lead obviously to the definition of C*-algebras and other operator algebras.

**Hilbert spaces**

The Hilbert spaces can completely categorize as follows. There is a unique Hilbert space up to isomorphism for each cardinality of the orthonormal basis. The Finite-dimensional Hilbert spaces are totally understood in linear algebra and infinite-dimensional separable Hilbert spaces are isomorphic too. Separability is being significant for applications; that is the functional analysis of Hilbert spaces consequently deals with this space. The one open problem in functional analysis is to prove that every bounded linear operator on a Hilbert space has an appropriate invariant subspace. Different special cases of this invariant subspace problem have already been proven.

**Banach spaces**

Banach spaces are more complicated than Hilbert spaces and cannot be classified in such a simple manner as those. Particularly, Banach spaces lack a notion analogous to an orthonormal basis.

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