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

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

if $1\,\leq p\,<\infty$ , or

$\mathop {{\rm {ess}}\sup } _{z\in \mathbb {T} }|f(z)|<\infty ,$ if $p\,=\,\infty$ . Then $L^{p}(\mathbb {T} )$ is a Banach space with a norm $\|\cdot \|_{p}$ defined by

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

if $1\,\leq \,p<\infty$ , or

$\|f\|_{\infty }=\mathop {{\rm {ess}}\sup } _{z\in \mathbb {T} }|f(z)|,$ if $p\,=\,\infty$ . The case p = 2 is special since it is also a Hilbert space and is in fact the only Hilbert space among the $L^{p}(\mathbb {T} )$ spaces, $1\,\leq p\,\leq \infty$ .