# Banach space

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

In mathematics, particularly in the branch known as functional analysis, a Banach space is a complete normed space. It is named after famed Hungarian-Polish mathematician Stefan Banach.

The space of all continous complex (resp. real) linear functionals of a complex (resp. real) Banach space is called its dual space. This dual space is also a Banach space when endowed with the operator norm on the continuous (hence, bounded) linear functionals.

## Examples of Banach spaces

1. The Euclidean space  with any norm is a Banach space. More generally, any finite dimensional normed space is a Banach space (due to its isomorphism to some Euclidean space).

2. Let , , denote the space of all complex-valued measurable functions on the unit circle  of the complex plane (with respect to the Haar measure  on ) satisfying:

,

if , or



if . Then  is a Banach space with a norm  defined by

,

if , or



if . The case p = 2 is special since it is also a Hilbert space and is in fact the only Hilbert space among the  spaces, .