Information AboutLp Space |
| CATEGORIES ABOUT LP SPACE | |
| normed spaces | |
| mathematical series | |
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In Mathematics , the ''Lp'' and ''' spaces''' are spaces of '' P-power Integrable Functions '', and corresponding '' Sequence Space s''. They form an important class of examples of Banach Space s in Functional Analysis , and of Topological Vector Space s. ''Lp'' spaces have applications in physics, statistics, finance, engineering, etc. MOTIVATION Consider the Real Vector Space R''n''. The sum of vectors in R''n'' is given by : and the scalar action is given by : The length of a vector is usually given by | ||
| :<math>\ \x\ P | \left(x_1^p+x_2^p+\dots+x_n^p
ight)^{1/p}</math> |
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| :<math>\ \x\ P | \left(x_1^p+x_2^p+\dots+x_n^p+x_{n+1}^p+\dots
ight)^{1/p}</math> |
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| :<math>\ \x\ \infty | \sup(x_1, x_2, \dots, x_n,x_{n+1}, \dots)</math> |
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| :<math>\ \x\ \infty | \lim_{p o\infty}\x\_p</math> |
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| The Space <math>\ell^2</math> Is The Only <math>\ell^p</math> Space That Is A | "http://wwwinformationdelightinfo/information/entry/Hilbert_space" class="copylinks">Hilbert Space , since any norm that is induced by an inner product should satisfy the parallelogram identity <math>\x+y\_p^2 + \x-y\_p^2= 2\x\_p^2 + 2\y\_p^2</math> Direct substitution with unit vectors results in a counter example |
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| The Dual Of ''c''<sub>0</sub> Is <math>\ell^1</math> The Dual Of <math>\ell^1</math> Is <math>\ell^\infty</math> For The Case Of Natural Numbers Index Set, The <math>\ell^p</math> And ''c''<sub>0</sub> Are | "http://wwwinformationdelightinfo/information/entry/separable_space" class="copylinks">Separable , with the sole exception of <math>\ell^{\,\infty}</math> Here, ''c''<sub>0</sub> is defined as the space of all sequences converging zero, with norm identical to ''x''<sub>&infin</sub> |
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| :<math>\f\ P : | \left({\int f^p\\mathrm{d}\mu}
ight)^{1/p}<\infty </math> |
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| Thus The Set Of ''p''<sup>th</sup> Power Integrable Functions, Together With The Function ·<sub>''p''</sub>, A | "http://wwwinformationdelightinfo/information/entry/seminorm" class="copylinks">Seminorm ed vector space, which we denote by |
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| ·<sub>''p''</sub> Since ''f''<sub>''p''</sub> | 0 if and only if ''f'' = 0 Almost Everywhere , in the quotient space two functions ''f'' and ''g'' are identified if ''f'' = ''g'' almost everywhere The resulting normed vector space is, by definition, |
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| :<math>L^p(S, \mu) : | \mathcal{L}^p(S, \mu) / \mathrm{ker}(\\cdot\_p) </math> |
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| :<math>\f\ \infty : | \inf \{ C\ge 0 : f(x) \le C \mbox{ for almost every } x\}</math> |
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| :<math>\f\ \infty | \lim_{p o\infty}\f\_p</math> |
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| If 0 < ''p'' < 1, Then ''L<sup>p</sup>'' Can Be Defined As Above, But  · <sub>''p''</sub> Does Not Satisfy The Triangle Inequality In This Case, And Hence It Defines Only A | "http://wwwinformationdelightinfo/information/entry/quasi-norm" class="copylinks">Quasi-norm However, we can still define a Metric by setting ''d''(''f'', ''g'') = (''f'' &minus ''g''<sub>''p''</sub>)<sup>''p''</sup> The resulting metric space is Complete , and ''L<sup> p</sup>'' for 0 < ''p'' < 1 is the prototypical example of an F-space that is not Locally Convex |
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| :<math>\ \ U \ {L^{p} (S, W \, \mathrm{d} \mu)} : | \left( \int_{S} w(x) u(x) ^{p} \, \mathrm{d} \mu (x)
ight)^{1/p}</math> |
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