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L^infty-Space

The space called L^infty (ell-infinity) generalizes the L-p-spaces to p=infty. No integration is used to define them, and instead, the norm on L^infty is given by the essential supremum.

More precisely,

|f|_infty= ess sup |f|

is the norm which makes L^infty a Banach space. It is the space of all essentially bounded functions. The space of bounded continuous functions is not dense in L^infty.

See also

Banach Space, Completion, Dense, Essential Supremum, L-p-Space, L2-Space, Measure, Measurable Function, Measure Space

This entry contributed by Todd Rowland

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Cite this as:

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

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