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$L^p$-space (Definition)

Definition

Let $ (X, \mathfrak{B}, \mu)$ be a measure space. Let $ 0<p < \infty$. The $ L^{p}$-norm of a function $ f:X\rightarrow \mathbb{C}$ is defined as
$\displaystyle \left\vert\left\vert f\right\vert\right\vert _{p} :=\left(\int_{X} \left\vert f\right\vert^p d\mu \right)^{\frac{1}{p}}$ (1)

when the integral exists. The set of functions with finite $ L^{p}$-norm forms a vector space $ V$ with the usual pointwise addition and scalar multiplication of functions. In particular, the set of functions with zero $ L^{p}$-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 $ L^{p}$-norm, identified up to equivalence almost everywhere. This quotient space is the complex $ L^{p}$-space on $ X$.

Theorem

If $ 1 \leq p < \infty$, the vector space $ V/K$ is complete with respect to the $ L^{p}$ norm.

The space $ L^{\infty}$.

The space $ L^{\infty}$ is somewhat special, and may be defined without explicit reference to an integral. First, the $ L^{\infty}$-norm of $ f$ is defined to be the essential supremum of $ \left\vert f\right\vert$:
$\displaystyle \left\vert\left\vert f\right\vert\right\vert _{\infty} :=\mathrm{... ...n \mathbb{R}: \mu(\left\{x: \left\vert f(x)\right\vert > a\right\}) = 0\right\}$ (2)

The definitions of $ V$, $ K$, and $ L^{\infty}$ then proceed as above, and again we have that $ L^\infty$ is complete. Functions in $ L^{\infty}$ are also called essentially bounded.

Example

Let $ X = [0,1]$ and $ f(x) = \frac{1}{\sqrt{x}}$. Then $ f \in L^{1}(X)$ but $ f \notin L^{2}(X)$.



"$L^p$-space" is owned by Mathprof. [ full author list (5) | owner history (4) ]
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See Also: measure space, norm, essential supremum, measure, Feynman path integral, amenable group, vector p-norm, vector norm, Sobolev inequality, $L^2$-spaces are Hilbert spaces

Other names:  $L^p$ space, essentially bounded function
Also defines:  $p$-integrable function, $L^\infty$, essentially bounded, $L^p$-norm

Attachments:
locally integrable function (Definition) by matte
proof that $L^p$ spaces are complete (Proof) by Simone
$\ell^p(X)$ space (Definition) by asteroid
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Cross-references: definitions, essential supremum, 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 17 references to this entry.

This is version 24 of $L^p$-space, born on 2002-02-17, modified 2007-05-24.
Object id is 2047, canonical name is LpSpace.
Accessed 19982 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)
Pending Errata and Addenda
None.
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L^\infty special? by akrowne on 2002-02-17 17:13:11
I have a question... can't you just define L^\infty as a limit (in p)?
-apk
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