Editing Lp space (section)

===Vector-valued {{math|''L<sup>p</sup>''}} spaces===

Given a measure space <math>(\Omega, \Sigma, \mu)</math> and a [[Locally convex topological vector space|locally convex space]] <math>E</math> (here assumed to be [[Complete topological vector space|complete]]), it is possible to define spaces of <math>p</math>-integrable <math>E</math>-valued functions on <math>\Omega</math> in a number of ways. One way is to define the spaces of [[Bochner integral|Bochner integrable]] and [[Pettis integral|Pettis integrable]] functions, and then endow them with [[Locally convex topological vector space|locally convex]] [[Vector topology|TVS-topologies]] that are (each in their own way) a natural generalization of the usual <math>L^p</math> topology. Another way involves [[topological tensor product]]s of <math>L^p(\Omega, \Sigma, \mu)</math> with <math>E.</math> Element of the vector space <math>L^p(\Omega, \Sigma, \mu) \otimes E</math> are finite sums of simple tensors <math>f_1 \otimes e_1 + \cdots + f_n \otimes e_n,</math> where each simple tensor <math>f \times e</math> may be identified with the function <math>\Omega \to E</math> that sends <math>x \mapsto e f(x).</math> This [[tensor product]] <math>L^p(\Omega, \Sigma, \mu) \otimes E</math> is then endowed with a locally convex topology that turns it into a [[topological tensor product]], the most common of which are the [[projective tensor product]], denoted by <math>L^p(\Omega, \Sigma, \mu) \otimes_\pi E,</math> and the [[injective tensor product]], denoted by <math>L^p(\Omega, \Sigma, \mu) \otimes_\varepsilon E.</math> In general, neither of these space are complete so their [[Complete topological vector space|completions]] are constructed, which are respectively denoted by <math>L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\pi E</math> and <math>L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\varepsilon E</math> (this is analogous to how the space of scalar-valued [[simple function]]s on <math>\Omega,</math> when seminormed by any <math>\|\cdot\|_p,</math> is not complete so a completion is constructed which, after being quotiented by <math>\ker \|\cdot\|_p,</math> is isometrically isomorphic to the Banach space <math>L^p(\Omega, \mu)</math>). [[Alexander Grothendieck]] showed that when <math>E</math> is a [[nuclear space]] (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.