The Inequalities of Young, H"{o}lder, and Minkowski

Show that if Hölder's Inequality holds for normalized functions, then it holds in general.

For $p_{1} < p_{2}$ and $E = (0,1)$, define $f(x) = x^{\alpha }$ for $0 < x < 1$, where $-1 / p_{1} < \alpha \leq -1 / p_{2}$. Show that $f \in L^{p_{1}}(E) \sim L^{p_{2}}(E)$.

Define $f(x) = \frac{x^{-1 / 2}}{1+\left|\ln x\right|}$ for $x > 0$. Verify that $f \in L^{p}(0, \infty )$ if and only if $p = 2$.

Define $f(t) = t$ for $0 \leq t \leq 1$. Fix $1 \leq p < \infty$. Find the dual function of $f \in L^{p}[0,1]$. Observe that it depends on $p$.

Define $f(t) = t$ for $0 \leq t \leq 1$. Then $f \in L^{\infty }[0,1]$. Show that there is no function $f^{*} \in L^{1}[0,1]$ for which

Does this contradict (3)?

Let $f, g \in L^{2}(E)$. Show that for any number $\lambda$,

and from this and the quadratic formula directly derive the Cauchy-Schwarz Inequality.

For $1 < p < \infty , q$ the conjugate of $p$ and $f \in L^{p}(E)$, show that if $A \subseteq E$ has finite measure, then

Use this to conclude that a bounded subset of $L^{p}(E)$ is uniformly integrable.

Show that in Young's Inequality there is equality if and only if $a^{p} = b^{q}$.

Show that in Hölder's Inequality there is equality if and only if there are constants $\alpha$ and $\beta$, not both zero, for which

For $1 \leq p < \infty$ and $f \in L^{p}(E)$, show that for each $\lambda > 0$,

For $1 \leq p < \infty$, suppose that $\left\{ f_{n}\right\} \rightarrow f$ in $L^{p}(E)$. Show that $\left\{ f_{n}\right\} \rightarrow f$ in measure, and therefore there is a subsequence of $\left\{ f_{n}\right\}$ that converges pointwise almost everywhere on $E$ to $f$.

Assume that $m(E) < \infty$. For $f \in L^{\infty }(E)$, show that $\lim_{p \rightarrow \infty }\left\| f\right\|_{p} = \left\| f\right\|_{\infty }$.

For $1 \leq p < \infty , q$ the conjugate of $p$, and $f \in L^{p}(E)$, show that

For $1 \leq p < \infty$ and $f \in L^{p}(E)$, show that $f = 0$ if and only if $\int_{E} f \cdot \psi d m = 0$ for every finitely supported, simple function $\psi : E \rightarrow \mathbb {R}$.

For $1 \leq p < \infty , q$ the conjugate of $p$ and $f \in L^{p}(E)$, show that $f = 0$ if and only if $\int_{E} f \cdot g d m = 0$ for $g \in L^{q}(E)$.

For $1 \leq p < \infty$, find the values of the parameter $\lambda$ for which

Let $1 \leq p \leq \infty$ and $q$ be the conjugate of $p$. If $\left\{ f_{n}\right\} \rightarrow f$ in $L^{p}(E)$ and $\left\{ g_{n}\right\} \rightarrow g$ in $L^{q}(E)$, show that

Assume that $m(E) < \infty$ and $1 \leq p_{1} < p_{2} \leq \infty$. Show that if $\left\{ f_{n}\right\} \rightarrow f$ in $L^{p_{2}}(E)$, then $\left\{ f_{n}\right\} \rightarrow f$ in $L^{p_{1}}(E)$.

Let $\left\{ f_{n}: E \rightarrow \mathbb {R}\right\}$ be a sequence of measurable functions that converges pointwise almost everywhere on $E$ to $f$. For $1 \leq p < \infty$, suppose that there is a function $g$ in $L^{p}(E)$ such that for all $n,\left|f_{n}\right| \leq g$ almost everywhere on $E$. Prove that $\left\{ f_{n}\right\} \rightarrow f$ in $L^{p}(E)$.

For $1 \leq p < \infty$ and $f \in L^{p}(\mathbb {R})$, show that $\left\{ f \cdot \chi_{[-n, n]}\right\} \rightarrow f$ in $L^{p}(\mathbb {R})$.

Assume that $m(E) < \infty$ and $1 \leq p < \infty$. Suppose that $\left\{ f_{n}: E \rightarrow \mathbb {R}\right\}$ is a sequence of measurable functions that converges pointwise almost everywhere on $E$ to $f$. For $1 \leq p < \infty$, show that $\left\{ f_{n}\right\} \rightarrow f$ in $L^{p}(E)$ if there is a $\theta > 0$ for which $\left\{ f_{n}\right\}$ is a bounded subset of $L^{p+\theta }(E)$.

For $1 \leq p < \infty$, show that if the absolutely continuous function $f$ on $[a, b]$ is the indefinite integral of an $L^{p}[a, b]$ function, then there is an $M > 0$ such that for any partition $\left\{ x_{0}, \ldots , x_{n}\right\}$ of $[a, b]$,