The set of -functions generalizes L2-space. Instead of square integrable, the measurable function must be -integrable for to be in .

On a measure space , the norm of a function is

The -functions are the functions for which this integral converges. For , the space of -functions is a Banach space which is not a Hilbert space.

The -space on , and in most other cases, is the completion of the continuous functions with compact support using the norm. As in the case of an L2-space, an -function is really an equivalence class of functions which agree almost everywhere. It is possible for a sequence of functions to converge in but not in for some other , e.g., converges in but not . However, if a sequence converges in and in , then its limit must be the same in both spaces.

For , the dual space to is given by integrating against functions in , where . This makes sense because of Hölder's inequality for integrals. In particular, the only -space which is self-dual is .

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

This entry contributed by Todd Rowland

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