$L^p$-space

Definition

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

$${||f||}_p:={\left({\int }_X{\left|f\right|}^p𝑑\mu \right)}^{\frac{1}{p}}$$(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 scalarmultiplication of functions. In particular, the set of functions with zero $L^p$-norm form a linear subspace of $V$, which for this articlewill 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 $X$.

Theorem

If , the vector space $V/K$ is complete with respect to the $L^p$ norm.

The space $L^{\mathrm{\infty }}$.

The space $L^{\mathrm{\infty }}$ is somewhat special, and may be defined without explicit reference to an integral. First, the $L^{\mathrm{\infty }}$-norm of $f$ isdefined to be the essential supremum of $\left|f\right|$:

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

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

The definitions of $V$, $K$, and $L^{\mathrm{\infty }}$ then proceed as above, and again we have that $L^{\mathrm{\infty }}$ is complete. Functions in $L^{\mathrm{\infty }}$ are also called essentially bounded.

Example

Let $X=[0,1]$ and $f(x)=\frac{1}{\sqrt{x}}$. Then $f\in L^1(X)$ but $f\notin L^2(X)$.