Banach space: Difference between revisions

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1. The Euclidean space <math>\scriptstyle \mathbb{R}^n</math> with any [[norm (mathematics)|norm]] is a Banach space. More generally, any finite dimensional normed space is a Banach space (due to its isomorphism to some Euclidean space).
1. The Euclidean space <math>\scriptstyle \mathbb{R}^n</math> with any [[norm (mathematics)|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 <math>\scriptstyle L^p(\mathbb{T})</math>, <math>\scriptstyle 1\, \leq p \,\leq\, \infty</math>, denote the space of all [[complex number|complex]]-valued measurable function on the unit circle <math>\scriptstyle \mathbb{T}\,=\,\{z \in \mathbb{C} \mid |z|\,=\,1\}</math> of the complex plane (with respect to the [[Haar measure]] <math>\scriptstyle \mu</math> on <math>\scriptstyle \mathbb{T}</math>) satisfying:     
2. Let <math>\scriptstyle L^p(\mathbb{T})</math>, <math>\scriptstyle 1\, \leq p \,\leq\, \infty</math>, denote the space of all [[complex number|complex]]-valued measurable functions on the unit circle <math>\scriptstyle \mathbb{T}\,=\,\{z \in \mathbb{C} \mid |z|\,=\,1\}</math> of the complex plane (with respect to the [[Haar measure]] <math>\scriptstyle \mu</math> on <math>\scriptstyle \mathbb{T}</math>) satisfying:     
:<math> \int_{\mathbb{T}}|f(z)|^p\,\mu(dz)<\infty</math>,
:<math> \int_{\mathbb{T}}|f(z)|^p\,\mu(dz)<\infty</math>,


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if <math>\scriptstyle p\,=\,\infty</math>. The case ''p'' = 2 is special since it is also a [[Hilbert space]] and is in fact the only Hilbert space among the <math>\scriptstyle L^p(\mathbb{T})</math> spaces, <math> \scriptstyle 1\,\leq p\,\leq \infty</math>.
if <math>\scriptstyle p\,=\,\infty</math>. The case ''p'' = 2 is special since it is also a [[Hilbert space]] and is in fact the only Hilbert space among the <math>\scriptstyle L^p(\mathbb{T})</math> spaces, <math> \scriptstyle 1\,\leq p\,\leq \infty</math>.
==Further reading==
1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980

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This editable Main Article is under development and subject to a disclaimer.

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 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 , , denote the space of all complex-valued measurable functions on the unit circle of the complex plane (with respect to the Haar measure on ) satisfying:

,

if , or

if . Then is a Banach space with a norm defined by

,

if , or

if . The case p = 2 is special since it is also a Hilbert space and is in fact the only Hilbert space among the spaces, .