Let (X,𝔅,μ) be a measure spaceMathworldPlanetmath. Let 0<p<. The Lp-norm of a function f:X is defined as

||f||p:=(X|f|p𝑑μ)1p (1)

when the integral exists. The set of functions with finite Lp-norm forms a vector spaceMathworldPlanetmath 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 1p<, the vector space V/K is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with respect to the Lp norm.

The space L.

The space L is somewhat special, and may be defined without explicit reference to an integral. First, the L-norm of f is defined to be the essential supremumMathworldPlanetmath of |f|:

||f||:=esssup|f|=inf{a:μ({x:|f(x)|>a})=0} (2)

However, if μ is the trivial measure, then essential supremum of every measurable functionMathworldPlanetmath is defined to be 0.

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


Let X=[0,1] and f(x)=1x. Then fL1(X) but fL2(X).

Title Lp-space
Canonical name Lpspace
Date of creation 2013-03-22 12:21:32
Last modified on 2013-03-22 12:21:32
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 28
Author Mathprof (13753)
Entry type Definition
Classification msc 28B15
Synonym Lp space
Synonym essentially bounded function
Related topic MeasureSpace
Related topic Norm
Related topic EssentialSupremum
Related topic Measure
Related topic FeynmannPathIntegral
Related topic AmenableGroup
Related topic VectorPnorm
Related topic VectorNorm
Related topic SobolevInequality
Related topic L2SpacesAreHilbertSpaces
Related topic RieszFischerTheorem
Related topic BoundedLinearFunctionalsOnLpmu
Related topic ConvolutionsOfComplexFunctionsOnLocallyCompactG
Defines p-integrable function
Defines L
Defines essentially bounded
Defines Lp-norm