Spaces of Integrable Functions

Prove that for any measurable functions on a set $X$ with measure $\mu$ the Hölder inequality is valid:

where the numbers $p$ and $q$ are connected by the relation $1/p+1/q=1$.

Prove that

for any measurable function $f$ on a set $X$ with measure $\mu$, where the supremum is taken over all functions $g(x)$ satisfying the inequality $\int_X\left|g(x)\right|^q\, d\mu (x)\leq 1$, and the numbers $p$ and $q$ are connected by the relation $1/p+1/q=1$.

Prove the Minkowski integral inequality

for $1<p<\infty$.

Say that a measure $\mu$ on a set $X$ has a countable base if there exists a countable family $\left\{ A_n\right\}$ of measurable subsets of $X$ such that for any measurable subset $B$ there is a set $A_n$ for which $\mu (A_n\mathbin {\triangle }B)<\varepsilon$.

Prove that $L^{1}\left(X,\mu \right)$ is a separable space if and only if $\mu$ has a countable base.

Prove that $L^{p}\left(X,\mu \right)$, $1<p<\infty$, is a separable space if and only if $L^{1}\left(X,\mu \right)$ is.

Prove that $L^{\infty }\left(X,\mu \right)$ is either finite-dimensional or nonseparable.

Let $\mu (X)<\infty$. Prove that for $p\geq q\geq 1$ the space $L^{p}\left(X,\mu \right)$ is contained in the space $L^{q}\left(X,\mu \right)$.

Prove that if $p\neq q$, then neither of the spaces $L^{p}\left(\mathbb {R},dx\right)$, $L^{q}\left(\mathbb {R},dx\right)$ is contained in the other.

Let $0<\alpha \leq \beta <\infty$. For what $p$ does the function $f(x)=1/(x^\alpha +x^\beta )$ belong to $L^{p}\left(\mathbb {R}_+,dx\right)$ ($\mathbb {R}_+$ denotes the positive half-line)?

Suppose that the numbers $p$, $q$, $r$ are connected by the relation $1/p+1/q+1/r=1$, and that $f\inL^{p}\left(X,\mu \right)$, $g\inL^{q}\left(X,\mu \right)$, $h\inL^{r}\left(X,\mu \right)$. Prove that $fgh$ is an integrable function and that $\left\| fgh\right\|_{1}\leq \left\| f\right\|_{p}\left\| g\right\|_{q}\left\| h\right\|_{r}$.

Let $1\leq p\leq r\leq q\leq \infty$. Prove that $L^{p}\left(X,\mu \right)\capL^{q}\left(X,\mu \right)$ is contained in $L^{r}\left(X,\mu \right)$, and that $\left\| f\right\|_{r}\leq \left\| f\right\|_{p}^\alpha \left\| f\right\|_{q}^\beta$ for every function $f\inL^{p}\left(X,\mu \right)\capL^{q}\left(X,\mu \right)$, where $\alpha =(r^{-1}-q^{-1})/(p^{-1}-q^{-1})$, $\beta =(p^{-1}-r^{-1})/(p^{-1}-q^{-1})$.

Let $\mu (X)<\infty$. Prove that $L^{\infty }\left(X,\mu \right)\subsetL^{p}\left(X,\mu \right)$ for all $p\geq 1$, and that $\left\| f\right\|_{\infty }=\lim_{p\to \infty }\left\| f\right\|_{p}$.

Prove that the following sets of functions are dense in $L^{p}\left({[a,b]}\right)$:

Find the norm of the function $f(x)=x^\alpha$ in those spaces $L^{p}\left({[0,1]}\right)$, $1\leq p\leq \infty$, to which it belongs.

In $L^{1}\left(\mathbb {R},dx\right)$ construct:

Let $\mu (X)<\infty$ and suppose that $V\subsetL^{1}\left(X,\mu \right)$ is a closed infinite-dimensional subspace. Prove that $V$ cannot be contained in $L^{\infty }\left(X,\mu \right)$.

Prove that the set $C_0(\mathbb {R})$ of continuous compactly supported functions is dense in $L^{p}\left(\mathbb {R},dx\right)$ for $1\leq p<\infty$.

Prove that every function $f\inL^{p}\left(\mathbb {R},dx\right)$ is continuous in the mean, i.e., for any $\varepsilon >0$ there is a $\delta >0$ such that for $\left|t\right|<\delta$

Prove the assertion of Problem 386 for the space $L^{p}\left(\mathbb {R}^{n},dx\right)$.

Let $M\subsetL^{p}\left(\mathbb {R}^{n},dx\right)$, $1\leq p<\infty$. Prove that $M$ is pre-compact if and only if:

1. There is a constant $c$ such that $\left\| f\right\|_{p}\leq c$ for all $f\in M$;

2. For any $\varepsilon >0$ there is a number $R(\varepsilon )$ such that

   $$\int _{\left|x\right|>R(\varepsilon )}\left|f(x)\right|\, dx<\varepsilon ,\qquad f\in M;$$

3. For any $\varepsilon >0$ there is a number $\delta (\varepsilon )>0$ such that for $\left|t\right|<\delta (\varepsilon )$

   $$\int _{\mathbb {R}^{n}}\left|f(x+t)-f(x)\right|^p\, dx<\varepsilon .$$

Verify the isomorphism of spaces

Find the extreme points of the unit ball in $L^{p}\left(X,\mu \right)$:

Prove that the spaces $L^{1}\left({[0,1]}\right)$ and $l_1$ are not isomorphic.