Banach space

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This editable Main Article is under development and subject to a disclaimer.

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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[edit]

1\. The Euclidean space ℝn 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 Lp(𝕋), 1≤p≤∞, denote the space of all complex-valued measurable functions on the unit circle 𝕋={z∈ℂ∣|z|=1} of the complex plane (with respect to the Haar measure μ on 𝕋) satisfying:

∫𝕋|f(z)|pμ(dz)<∞,

if 1≤p<∞, or

esssupz∈𝕋|f(z)|<∞,

if p=∞. Then Lp(𝕋) is a Banach space with a norm ‖⋅‖p defined by

‖f‖p=(∫𝕋|f(z)|pμ(dz))1/p,

if 1≤p<∞, or

‖f‖∞=esssupz∈𝕋|f(z)|,

if p=∞. The case p = 2 is special since it is also a Hilbert space and is in fact the only Hilbert space among the Lp(𝕋) spaces, 1≤p≤∞.