Banach space: Difference between revisions

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imported>Hendra I. Nurdin
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if <math>\scriptstyle p\,=\,\infty</math>. Then <math>\scriptstyle L^p(\mathbb{T})</math> is a Banach space with a norm <math>\scriptstyle \|\cdot \|_p</math> defined by
if <math>\scriptstyle p\,=\,\infty</math>. Then <math>\scriptstyle L^p(\mathbb{T})</math> is a Banach space with a norm <math>\scriptstyle \|\cdot \|_p</math> defined by


:<math> \|f\|_p=\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)</math>,
:<math> \|f\|_p=\left(\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)\right^{1/p}</math>,


if <math>\scriptstyle 1\,\leq\, p < \infty </math>, or  
if <math>\scriptstyle 1\,\leq\, p < \infty </math>, or  

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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 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 function 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

Failed to parse (syntax error): {\displaystyle \|f\|_p=\left(\int_{\mathbb{T}}|f(z)|^p\,\mu(dz)\right^{1/p}} ,

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, .

Further reading

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980