143.215.126.200 at 21:17, 1 February 2014

2014-02-01T21:17:16Z

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	~~The writer~~ is ~~called Irwin~~. ~~California is our beginning location~~. ~~To gather cash~~ is 1 of the ~~issues I adore most~~. ~~Since she was eighteen she~~'s ~~been operating as~~ a ~~meter reader but she~~'~~s always needed her personal business~~.<br><br>~~Here~~ is ~~my site~~: ~~[http~~://~~www~~.~~article~~-~~galaxy~~.~~com~~/~~profile~~.~~php?~~a~~=165109 http~~://~~www~~.~~article~~-~~galaxy~~.~~com/profile~~.~~php?a~~=~~165109~~]		In [[mathematics]], '''Bochner spaces''' are a generalization of the concept of [[Lp space\|''L<sup>p</sup>'' spaces]] to functions whose values lie in a [[Banach space]] which is not necessarily the space R or C of real or complex numbers.

			The space ''L<sup>p</sup>(X)'' consists of (equivalence classes of) all [[Bochner measurable]] functions ''f'' with values in the Banach space ''X'' whose [[norm (mathematics)\|norm]] ''\|\|f\|\|<sub>X</sub>'' lies in the standard ''L<sup>p</sup>'' space. Thus, if ''X'' is the set of complex numbers, it is the standard Lebesgue ''L<sup>p</sup>'' space.

			Almost all standard results on ''L<sup>p</sup>'' spaces do hold on Bochner spaces too; in particular, the Bochner spaces ''L<sup>p</sup>(X)'' are Banach spaces for <math>1\le p\le \infty</math>.

			==Background==
			Bochner spaces are named for the [[Poland\|Polish]]-[[United States\|American]] [[mathematician]] [[Salomon Bochner]].

			==Applications==
			Bochner spaces are often used in the [[functional analysis]] approach to the study of [[partial differential equation]]s that depend on time, e.g. the [[heat equation]]: if the temperature <math>g(t,x)</math> is a scalar function of time and space, one can write <math>(f(t))(x):=g(t,x)</math> to make ''f'' a family ''f(t)'' (parametrized by time) of functions of space, possibly in some Bochner space.

			==Definition==

			Given a [[measure space]] (''T'', Σ, ''μ''), a [[Banach space]] (''X'', \|\| · \|\|<sub>''X''</sub>) and 1 ≤ ''p'' ≤ +∞, the '''Bochner space''' ''L''<sup>''p''</sup>(''T''; ''X'') is defined to be the [[Kolmogorov quotient]] (by equality [[almost everywhere]]) of the space of all [[Bochner measurable]] functions ''u'' : ''T'' → ''X'' such that the corresponding norm is finite:

			:<math>\\| u \\|_{L^{p} (T; X)} := \left( \int_{T} \\| u(t) \\|_{X}^{p} \, \mathrm{d} \mu (t) \right)^{1/p} < + \infty \mbox{ for } 1 \leq p < \infty,</math>

			:<math>\\| u \\|_{L^{\infty} (T; X)} := \mathrm{ess\,sup}_{t \in T} \\| u(t) \\|_{X} < + \infty.</math>

			In other words, as is usual in the study of ''L''<sup>''p''</sup> spaces, ''L''<sup>''p''</sup>(''T''; ''X'') is a space of [[equivalence class]]es of functions, where two functions are defined to be equivalent if they are equal everywhere except upon a ''μ''-[[measure zero]] subset of ''T''. As is also usual in the study of such spaces, it is usual to [[abuse of notation\|abuse notation]] and speak of a "function" in ''L''<sup>''p''</sup>(''T''; ''X'') rather than an equivalence class (which would be more technically correct).

			==Application to PDE theory==

			Very often, the space ''T'' is an [[Interval (mathematics)\|interval]] of time over which we wish to solve some partial differential equation, and ''μ'' will be one-dimensional [[Lebesgue measure]]. The idea is to regard a function of time and space as a collection of functions of space, this collection being parametrized by time. For example, in the solution of the heat equation on a region Ω in '''R'''<sup>''n''</sup> and an interval of time [0, ''T''], one seeks solutions
			:<math>u \in L^{2} \left( [0, T]; H_{0}^{1} (\Omega) \right)</math>
			with time derivative
			:<math>\frac{\partial u}{\partial t} \in L^{2} \left( [0, T]; H^{- 1} (\Omega) \right).</math>
			Here <math>H_{0}^{1} (\Omega)</math> denotes the [[Sobolev space\|Sobolev]] [[Hilbert space]] of once-[[weak derivative\|weakly differentiable]] functions with first weak derivative in ''L''²(Ω) that vanish at the [[boundary (topology)\|boundary]] of Ω (in the sense of trace, or, equivalently, are limits of smooth functions with [[compact space\|compact]] [[support (mathematics)\|support]] in Ω); <math>H^{-1} (\Omega)</math> denotes the [[dual space]] of <math>H_{0}^{1} (\Omega)</math>.

			(The "[[partial derivative]]" with respect to time ''t'' above is actually a [[total derivative]], since the use of Bochner spaces removes the space-dependence.)

			==References==

			* {{cite book \| last=Evans \| first=Lawrence C. \| title=Partial differential equations \| location=Providence, RI \| publisher=American Mathematical Society \| year=1998 \| isbn=0-8218-0772-2}}

			==See also==
			*[[Vector-valued functions]]

			{{Functional Analysis}}

			[[Category:Functional analysis]]
			[[Category:Partial differential equations]]
			[[Category:Sobolev spaces]]

en>Jingleihu at 08:55, 19 August 2011

2011-08-19T08:55:24Z

New page

The writer is called Irwin. California is our beginning location. To gather cash is 1 of the issues I adore most. Since she was eighteen she's been operating as a meter reader but she's always needed her personal business.<br><br>Here is my site: [http://www.article-galaxy.com/profile.php?a=165109 http://www.article-galaxy.com/profile.php?a=165109]

Binding constant - Revision history

143.215.126.200 at 21:17, 1 February 2014

en>Jingleihu at 08:55, 19 August 2011