# A report onFunctional 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 analysis

## Hilbert space

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.

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

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

Polish 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

In 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.

## Hahn–Banach theorem

The Hahn–Banach theorem is a central tool in functional analysis.

## Banach space

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

## Function space

Set 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

Branch 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.