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 $\scriptstyle \mathbb{R}^n$ 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, $\scriptstyle 1\,\leq p\,\leq \infty$ .