The $L^{p}$ Spaces: Completeness and Approximation

If $K \subseteq \mathbb {R}$ is compact, show that $\left\| \cdot \right\|_{\max }$ is a norm on $C(K)$.

For $f \in C[a, b]$, define $\left\| f\right\|_{1} = \int_{a}^{b}\left|f\right| d m$. Show that this is a norm on $C[a, b]$. Also show that there is a number $c \geq 0$ for which

but there is no $c \geq 0$ for which

Let $X$ be the linear space of all polynomials with real coefficients defined on $\mathbb {R}$. For a polynomial $p$, define $\left\| p\right\|$ to be the sum of the absolute values of the coefficients. Is this a norm?

For $f \in L^{1}[a, b]$, define $\left\| f\right\| = \int_{a}^{b} x^{2}\left|f(x)\right| d x$. Show that this is a norm on $L^{1}[a, b]$.

For $f \in L^{\infty }(E)$, show that $\left\| f\right\|_{\infty }$ is the smallest essential upper bound for $f$ and

For $0 < p < 1$, define $L^{p}(E)$ to be the collection of measurable functions $f: E \rightarrow \overline{\mathbb {R}}$ for which $\int_{E}\left|f\right|^{p} d m < \infty$. Show that $L^{p}(E)$ is a linear space.

Show that in a normed linear space, a convergent sequence is bounded.

State and prove the linearity property of convergent sequences in a normed linear space.

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 $[a, b]$,

Prove that $L^{\infty }(E)$ is complete.

Provide an example of a Cauchy sequence of real numbers that is not rapidly Cauchy.

Show that a normed linear space is complete if and only if each rapidly Cauchy sequence converges.

For each $n$, define $f_{n}(x) = x^{n}$ for $0 \leq x \leq 1$. Is $\left\{ f_{n}:[0,1] \rightarrow \mathbb {R}\right\}$ rapidly Cauchy in $C[0,1]$ ? For $1 \leq p < \infty$, is $\left\{ f_{n}\right\}$ rapidly Cauchy in $L^{p}([0,1])$ ?

Consider the linear space of polynomials restricted to $[a, b]$, normed by the maximum norm. Is this space complete?

Let $m(E) < \infty$ and $1 \leq p_{1} < p_{2} < \infty$. Consider the linear space $L^{p_{2}}(E)$ normed by $\left\| \cdot \right\|_{p_{1}}$. Is this normed linear space compete?

A subspace $X_{0}$ of a normed linear space $X$ is said to be closed provided that the limit of a convergent sequence in $X_{0}$ also belongs to $X_{0}$. Assume that $X$ is complete. Show that a subspace $X_{0}$ of $X$, with the norm it inherits from $X$, is complete if and only if it is closed.

Let $A \subseteq B \subseteq X$. Show that $A$ is dense in $B$ if and only if each $f \in B$ is the limit of a sequence $\left\{ f_{n}\right\}$ in $A$.

Let $\mathcal{F}$ be a dense subset of $X$ and suppose that $\mathcal{S} \subseteq X$ has the property that each $f \in \mathcal{F}$ is the limit of a sequence $\left\{ f_{n}\right\}$ in $\mathcal{S}$. Show that $\mathcal{S}$ is dense in $X$. In particular, show that if $\mathcal{S}$ is countable, then $X$ is separable.

Show that, for $1 \leq p < \infty , \ell^{p}$ is separable, but $\ell^{\infty }$ is not separable.

Show that the subspace of simple functions is dense in $L^{\infty }(E)$.

Show that $C[a, b]$ is not a dense subspace of $L^{\infty }[a, b]$.

Let $1 \leq p < \infty , q$ be the conjugate of $p$, and $\mathcal{S}$ be a dense subset of $L^{q}(E)$. Show that if $g \in L^{p}(E)$ and $\int_{E} f \cdot g d m = 0$ for all $f \in \mathcal{S}$, then $g = 0$.

Exhibit a measurable set $E$ for which $L^{\infty }(E)$ is separable. Show that $L^{\infty }(E)$ is not separable if the set $E$ contains a non-degenerate interval.