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[edit]

1. The Euclidean space ${\displaystyle {\mathbb{ℝ}}^n}$ 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 ${\displaystyle L^p(\mathbb{𝕋})}$, ${\displaystyle 1\le p\le \mathrm{\infty }}$, denote the space of all complex-valued measurable functions on the unit circle ${\displaystyle \mathbb{𝕋}=\{z\in \mathbb{ℂ}\mid |z|=1\}}$ of the complex plane (with respect to the Haar measure ${\displaystyle \mu }$ on ${\displaystyle \mathbb{𝕋}}$) satisfying:

${\displaystyle \underset{\mathbb{𝕋}}{\int }|f(z){|}^p\mu (dz)<\mathrm{\infty }}$,

if ${\displaystyle 1\le p<\mathrm{\infty }}$, or

${\displaystyle {\mathrm{e}\mathrm{s}\mathrm{s}\sup }_{z\in \mathbb{𝕋}}|f(z)|<\mathrm{\infty },}$

if ${\displaystyle p=\mathrm{\infty }}$. Then ${\displaystyle L^p(\mathbb{𝕋})}$ is a Banach space with a norm ${\displaystyle \Vert \cdot {\Vert }_p}$ defined by

${\displaystyle \Vert f{\Vert }_p={(\underset{\mathbb{𝕋}}{\int }|f(z){|}^p\mu (dz))}^{1/p}}$,

if ${\displaystyle 1\le p<\mathrm{\infty }}$, or

${\displaystyle \Vert f{\Vert }_{\mathrm{\infty }}={\mathrm{e}\mathrm{s}\mathrm{s}\sup }_{z\in \mathbb{𝕋}}|f(z)|,}$

if ${\displaystyle p=\mathrm{\infty }}$. The case p = 2 is special since it is also a Hilbert space and is in fact the only Hilbert space among the ${\displaystyle L^p(\mathbb{𝕋})}$ spaces, ${\displaystyle 1\le p\le \mathrm{\infty }}$.