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 -space
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(Definition)
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Let $(X, \borel, \mu)$ be a measure space. Let $0<p < \infty$ . The $\Lp$ -norm of a function $f:X\rightarrow \complexes$ is defined as \begin{equation} \norm{f}_{p} \defined \left(\int_{X} \abs{f}^p d\mu \right)^{\frac{1}{p}} \end{equation}when the integral exists. The set of functions with finite $\Lp$ -norm forms a vector
space $V$ with the usual pointwise addition and scalar multiplication of functions. In particular, the set of functions with zero $\Lp$ -norm form a linear subspace of $V$ , which for this article will be called $K$ . We are then interested in the quotient space $V/K$ , which consists of complex functions on $X$ with finite $\Lp$ -norm, identified up to equivalence almost everywhere. This quotient space is the complex $\Lp$ -space on $X$ .
If $1 \leq p < \infty$ , the vector space $V/K$ is complete with respect to the $\Lp$ norm.
The space $\Linf$ is somewhat special, and may be defined without explicit reference to an integral. First, the $\Linf$ -norm of $f$ is defined to be the essential supremum of $\abs{f}$ : \begin{equation} \norm{f}_{\infty} \defined \esssup \abs{f} = \inf \set{a \in \reals: \mu(\set{x: \abs{f(x)} > a}) = 0} \end{equation}However, if $\mu$ is the trivial measure, then essential supremum of every measurable
function is defined to be 0.
The definitions of $V$ , $K$ , and $\Linf$ then proceed as above, and again we have that $L^\infty$ is complete. Functions in $\Linf$ are also called essentially bounded.
Let $X = [0,1]$ and $f(x) = \frac{1}{\sqrt{x}}$ . Then $f \in \Lone(X)$ but $f \notin \Ltwo(X)$ .
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See Also: measure space, norm, essential supremum, measure, Feynman path integral, amenable group, vector p-norm, vector norm, Sobolev inequality,  -spaces are Hilbert spaces, Riesz-Fischer theorem, bounded linear functionals on  ,  -norm is dual to 
| Other names: |
space, essentially bounded function |
| Also defines: |
 -integrable function,  , essentially bounded, -norm |
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Cross-references: definitions, measurable function, measure, essential supremum, reference, norm, complete, complex, almost everywhere, equivalence, complex functions, quotient space, linear subspace, multiplication, scalar, pointwise addition, vector space, finite, integral, function, measure space
There are 21 references to this entry.
This is version 25 of -space, born on 2002-02-17, modified 2008-12-24.
Object id is 2047, canonical name is LpSpace.
Accessed 25210 times total.
Classification:
| AMS MSC: | 28B15 (Measure and integration :: Set functions, measures and integrals with values in abstract spaces :: Set functions, measures and integrals with values in ordered spaces) |
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Pending Errata and Addenda
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