# $L^{p}$-space

## Definition

Let $(X,\mathfrak{B},\mu)$ be a measure space  . Let $0. The $L^{p}$-norm of a function $f:X\rightarrow\mathbb{C}$ is defined as

 $\left|\left|f\right|\right|_{p}:=\left(\int_{X}\left|f\right|^{p}d\mu\right)^{% \frac{1}{p}}$ (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 $X$.

## 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 $\left|f\right|$:

 $\left|\left|f\right|\right|_{\infty}:=\mathrm{ess\ sup}\left|f\right|=\inf% \left\{a\in\mathbb{R}:\mu(\left\{x:\left|f(x)\right|>a\right\})=0\right\}$ (2)

However, if $\mu$ is the trivial measure, then essential supremum of every measurable function  is defined to be 0.

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 $f\notin L^{2}(X)$.

Title $L^{p}$-space Lpspace 2013-03-22 12:21:32 2013-03-22 12:21:32 Mathprof (13753) Mathprof (13753) 28 Mathprof (13753) Definition msc 28B15 $L^{p}$ space essentially bounded function MeasureSpace Norm EssentialSupremum Measure FeynmannPathIntegral AmenableGroup VectorPnorm VectorNorm SobolevInequality L2SpacesAreHilbertSpaces RieszFischerTheorem BoundedLinearFunctionalsOnLpmu ConvolutionsOfComplexFunctionsOnLocallyCompactG $p$-integrable function $L^{\infty}$ essentially bounded $L^{p}$-norm