Infinite dimensional vector space

Definition 2 (TVS, nls, Banach space, ips, Hilbert space)  

Example 2 (Sequence space)   For $1\leq p\leq q\leq \infty$,

$$c_{00}\subsetneq, l^p\subsetneq l^q\subsetneq c_0 \subsetneq c\subsetneq l^\infty.$$

Example 3 (Function space)   Let $X$ be a compact set $(C(X), \Vert~\Vert _\infty)$, $(C^n(X), \Vert~\Vert _\infty)$

Definition 3 (Measure space)   Let $X$ be a set and $\Omega$ is a $\sigma$-algebra on $X$ and $\mu$ is a measure. Then for $1\leq p <\infty$,

$$L^p(X,\Omega, \mu):=\left\{f:X\rightarrow \mathbb{C} \middle\vert \int_X \vert f\vert^pd\mu < \infty \right\}$$

is a Banach space with respect to the norm, denoted by $\Vert~\Vert _p$, called as ``$p$-norm", and defined as

$$\Vert f\Vert _p =\left(\int_X \vert f\vert^pd\mu \right)^\frac{1}{p}.$$

For $p=\infty$ case,

$$L^\infty(X,\Omega, \mu):=\left\{f:X\r... ...arrow \mathbb{C} \middle\vert \inf\{c: \mu\{x\in X:f(x)>c\}=0 \}<\infty\right\}$$

is also a Banach space with respect to the norm, denoted by $\Vert~\Vert _\infty$, called as ``essential supremum", and defined as

$$\Vert f\Vert _\infty =\inf\Big\{c: \mu\{x:\vert f(x)\vert>c\}=0 \Big\}.$$

Note that those measurable functions, for which we can define the norm, are precisely collected in the appropriate space.

Theorem 2.1   Let $(X,\Omega, \mu)$ be a measure space and $\mu(X)$ is a finite positive number. Suppose $1< p < q < \infty$ and $f\in L^p(X,\Omega, \mu)$. Then

$$\frac{1}{\mu(X)^\frac{1}{p}}\Vert f\Vert _q\leq \frac{1}{\mu(X)^\frac{1}{q}}\Vert f\Vert _p$$

$$\Vert f\Vert _q\leq \mu(X)^{\left(\frac{1}{p}-\frac{1}{q}\right)}\Vert f\Vert _p$$

Proof. Use Holders inequality or Jensen's inequality with $\vert f\vert^p\in L^1$ and $1\in L^\infty$ with $p'=\frac{q}{p}>1$ $\qedsymbol$

From the above discussion, when $X$ has positive finite measure (ie $\mu(x)<\infty$), for $1 <p \leq q < \infty$,

$$L^\infty \subsetneq, L^p\subsetneq L^q\subsetneq L^1.$$

Remark 1   Among the above $L^p$ spaces, $L^2$ is a Hilbert space.