# A report on Functional analysis

Branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense.

- Functional analysis50 related topics with Alpha

## Hilbert space

14 linksIn mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.

The success of Hilbert space methods ushered in a very fruitful era for functional analysis.

## Vector space

10 links[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.

## Stefan Banach

8 linksPolish mathematician who is generally considered one of the 20th century's most important and influential mathematicians.

Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians.

He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics.

## Topological vector space

5 linksIn mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.

## Banach space

7 linksIn mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space.

## Function space

5 linksSet of functions between two fixed sets.

Set of functions between two fixed sets.

In functional analysis the same is seen for continuous linear transformations, including topologies on the vector spaces in the above, and many of the major examples are function spaces carrying a topology; the best known examples include Hilbert spaces and Banach spaces.

## Linear algebra

3 linksBranch of mathematics concerning linear equations such as:

Branch of mathematics concerning linear equations such as:

Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.

## Unitary operator

1 linksIn functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product.

## Mathematical analysis

3 linksBranch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

Branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.