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

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

if ${\displaystyle \scriptstyle 1\,\leq p\,<\infty }$, or

${\displaystyle \mathop {{\rm {ess}}\sup } _{z\in \mathbb {T} }|f(z)|<\infty ,}$

if ${\displaystyle \scriptstyle p\,=\,\infty }$. Then ${\displaystyle \scriptstyle L^{p}(\mathbb {T} )}$ is a Banach space with a norm ${\displaystyle \scriptstyle \|\cdot \|_{p}}$ defined by

${\displaystyle \|f\|_{p}=\left(\int _{\mathbb {T} }|f(z)|^{p}\,\mu (dz)\right)^{1/p}}$,

if ${\displaystyle \scriptstyle 1\,\leq \,p<\infty }$, or

${\displaystyle \|f\|_{\infty }=\mathop {{\rm {ess}}\sup } _{z\in \mathbb {T} }|f(z)|,}$

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