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L^p-Space


The set of L^p-functions (where p>=1) generalizes L2-space. Instead of square integrable, the measurable function f must be p-integrable for f to be in L^p.

On a measure space X, the L^p norm of a function f is

 |f|_(L^p)=(int_X|f|^p)^(1/p).

The L^p-functions are the functions for which this integral converges. For p!=2, the space of L^p-functions is a Banach space which is not a Hilbert space.

The L^p-space on R^n, and in most other cases, is the completion of the continuous functions with compact support using the L^p norm. As in the case of an L2-space, an L^p-function is really an equivalence class of functions which agree almost everywhere. It is possible for a sequence of functions f_n to converge in L^p but not in L^(p^') for some other p^', e.g., f_n=(1+x^2)^(-1/2-1/n) converges in L^2(R) but not L^1(R). However, if a sequence converges in L^p and in L^(p^'), then its limit must be the same in both spaces.

For p>1, the dual vector space to L^p is given by integrating against functions in L^q, where 1/p+1/q=1. This makes sense because of Hölder's inequality for integrals. In particular, the only L^p-space which is self-dual is L^2.

While the use of L^p functions is not as common as L^2, they are very important in analysis and partial differential equations. For instance, some operators are only bounded in L^p for some p>2.


See also

Banach Space, Completion, Hilbert Space, Lebesgue Integral, L-p-Function, L2-Space, Measure, Measure Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "L^p-Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Lp-Space.html

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