Banach space

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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, .