Lp Space

$L^p$ space for $1\le p<\infty$

Let$(X,\mathcal{M},\mu )$ be a measurable space then $L^p(\mu ):=\{f\text{measurable}:{\int }_X|f(x){|}^p\text{d}\mu (x)<\infty \}/\sim \mu -a.e.$ is a Banach Space $\forall 1\le p<\infty$. We consider the $L^p$ norm defined as $\Vert f{\Vert }_{L^p(X,\mathcal{M},\mu )}={\left({\int }_X|f(x){|}^{p}d\mu (x)\right)}^{\frac{1}{p}}$ and moreover, when $p=2$ it’s a Hilbert Space.

$L^{\infty }$

Let$(X,\mathcal{M},\mu )$ be a measurable space then $L^{\infty }(X,\mu ):=\{f:X\to \mathbb{R}\text{or}\mathbb{C}\text{measurable}:\exists E\mu (E)=0\text{s.t.}{\sup }_{x\in X\setminus E}|f(x)|<\infty \}/\sim \mu -a.e.$ We cam also define norm as $\Vert f{\Vert }_{L^{\infty }(X,\mu )}=\inf \{M>0:|f(x)|\le M\mu -a.e.x\in X\}$

Proposition

1. $C_c(X)$ denoting continuous functions with compact support is dense in $L^p(X)$ with $1\le p<\infty$. And it fails for $L^{\infty }$, think of $f=1\in L^{\infty }$
2. $C_c^{\infty }(X)$ denoting smooth functions with compact support is dense in $L^p(X)$ with $1\le p<\infty$. And it fails for $L^{\infty }$, think of $f=1\in L^{\infty }$
3. The set of simple functions is dense in $L^p$ for all $1\le p\le \infty$.
4. The set of indefinitely differentiable functions with compact support t is dense in $L^p(X)$ with $1\le p<\infty$.

Proposition

Consider $L^p=L^p({\mathbb{R}}^d)$ with Lebesgue measure. Let $f_0(x)=|x{|}^{-\alpha }$ if $|x|<1$, $f_0(x)=0$ for $|x|\ge 1$, also let $f_{\infty }(x)=|x{|}^{-\alpha }$ if $|x|\ge 1$, $f_{\infty }(x)=0$ when $|x|<1$. $f_0\in L^p$ if and only if $p\alpha <d$ $f_{\infty }\in L^p$ if and only if $d<p\alpha$.

Remark

Minkowski’s Inequality is just the triangle inequality in $L^p$ space. And Holder Inequality can be regarded as the extension of Cauchy Schwarz inequality.