Banach space

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

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

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

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

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

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

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

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

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