Completeness of $\LebesgueSpace{p}$

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $\left\{ f_{n}\right\}$ be a sequence in $L^{1}\left(X\right)$ that converges uniformly to a function $f$. Prove that $\left\{ f_{n}\right\}$ converges to $f$ in the norm of $L^{1}\left(X\right)$. Does this still hold if $\mu (X)=\infty$ ?

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and let $\left\{ f_{n}\right\}$ be a sequence in $L^{p}\left(X, \mu \right)$ that it is uniformly Cauchy. Prove that it converges to a function $f$ that belongs to $L^{p}$.

Prove the Riesz-Fischer theorem.

Theorem 10.3.3 (The Riesz-Fischer Theorem): Let $(X, \mathfrak {M}, \mu )$ be a measure space and $1 \leq p \leq \infty$. Then $L^{p}\left(X, \mu \right)$ is a Banach space. Moreover, if the sequence $\left\{ f_{n}\right\}$ converges to $f$ in $L^{p}\left(X, \mu \right)$, then there is a subsequence $\left\{ f_{n_{k}}\right\}$ that converges pointwise a.e. on $X$ to $f$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $\left\{ f_{n}\right\}_{n=1}^{\infty } \subset L^{p}\left(X, \mu \right), 1 \leq p \leq \infty$. Suppose that $\left\{ f_{n}\right\}$ converges to 0 in the norm of $L^{p}$. Prove that it converges to 0 in measure.

Let $(X, \mathfrak {M}, \mu )$ be a measure space, let $\left\{ f_{n}\right\}_{n=1}^{\infty }$ a sequence of measurable functions that converges in measure to a function $f$, and let $1 \leq p<\infty$.

Give an example of a sequence that converges in measure but not in the norm of $L^{p}\left(X, \mu \right), 1 \leq p \leq \infty$.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and let $\left\{ f_{n}\right\}$ be a sequence of nonnegative integrable functions such that $\lim \int_{X} f_{n} d \mu =0$. Prove or disprove: $\lim f_{n}=0$ a.e.

Give an example of a sequence $\left\{ f_{n}\right\}$ in $L^{1}\left(0,1\right)$ that has the following properties:

1. $\lim f_{n}(x)=0$ for all $x \in [0,1]$;

2. there is $C>0$ such that $\int_{0}^{1}\left|f_{n}\right| \leq C$, for all $n \in \mathbb {N}$;

3. $\left\{ f_{n}\right\}$ does not converge in the norm of $L^{1}\left(0,1\right)$.

Give an example to show that Theorem 10.3.5 is false when $p=\infty$.

Theorem 10.3.5: Let $(X, \mathfrak {M}, \mu )$ be a measure space and $1 \leq p < \infty$. Suppose that $\left\{ f_{n}\right\}$ is a sequence in $L^{p}\left(X, \mu \right)$ that converges a.e. on $X$ to a function $f$ that belongs to $L^{p}\left(X, \mu \right)$. Then $\lim_{n} \left\| f_{n}-f\right\|_{p} = 0$ if and only if $\lim_{n} \left\| f_{n}\right\|_{p} = \left\| f\right\|_{p}$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space and let $\left\{ f_{n}\right\} \in L^{p}\left(X, \mu \right) \cap L^{r}\left(X, \mu \right)$, for some $p, r \in [1,+\infty )$. Suppose that $\left\{ f_{n}\right\}$ converges to a function $g$ in the norm of $L^{p}\left(X, \mu \right)$ and to a function $h$ in the norm of $L^{r}\left(X, \mu \right)$. Prove or disprove: $g=h$ a.e.

Let

Given $p \in [1, \infty )$, define

Show that $\left\| f\right\|_{p}^{i n f}$ is a norm for each $p$, and that $X$ is a Banach space with each of these norms.

Let

Given $p_{1}, p_{2} \in [1, \infty )$, define

Show that $\left\| f\right\|_{p_{1}, p_{2}}$ is a norm for each pair $p_{1}, p_{2}$, and that $X$ is a Banach space with each of these norms.

Let $X=\mathbb {R}^{n+1}$ with the product measure $m_{n} \times m$ on the $\sigma$-algebra $\mathcal{M}_{n} \otimes \mathcal{M}$, and let $L^{p, r}$ be the space of equivalence classes of jointly measurable functions $f(x, t), x \in \mathbb {R}^{n}, t \in \mathbb {R}$, for which the norm is defined by

Prove that $\left\| f\right\|_{p, r}$ is a norm for each pair $1 \leq p, r<\infty$ and that with this norm $L^{p, r}$ is a Banach space.

Let $1 \leq p \leq r \leq q<\infty$. Prove that $L^{p}\left(0,1\right) \cap L^{q}\left(0,1\right)$ is a dense subspace of $L^{r}\left(0,1\right)$. Does the result hold if $q=\infty$ ?

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p<\infty$, and let $\left\{ f_{n}\right\} \in L^{p}\left(X, \mu \right)$. Suppose

Let $\left\{ a_{k}\right\}$ be a sequence of real numbers such that $\sum_{k}\left|a_{k}\right|^{p}<\infty$ and let $f=\sum_{k} a_{k} f_{k}$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p \leq \infty$, and let $\left\{ f_{n}\right\} \in L^{p}\left(X, \mu \right)$ such that $\lim_{n}\left\| f_{n}\right\|_{p}=0$. Prove that there exists a subsequence $\left\{ f_{n_{k}}\right\}$ and a nonnegative function $h \in L^{p}\left(X, \mu \right)$ such that $\left|f_{n_{k}}\right| \leq h$ a.e. $[\mu ]$ and $\lim_{k} f_{n_{k}}=0$ a.e. $[\mu ]$.

Give an example of a sequence $\left\{ f_{n}\right\}$ in $L^{p}\left(0,1\right)$ that converges to a function $f$ in the norm of $L^{p}\left(0,1\right)$ but not pointwise at any point of $[0,1]$.

Let $1 \leq p<\infty$, and let $E_{p}=\left\{ f \in L^{p}\left(\mathbb {R}\right): \int_{\mathbb {R}} f d \mu =0\right\}$. Prove that $E_{p}$ is closed if and only if $p=1$.

Let $A=\left\{ f \in L^{2}\left(0,1\right): \exists \text{ interval } I_{f} \subset [0,1], 1 / 2 \in I_{f}, f=0 \text{ a.e. on } I_{f}\right\}$. Prove or disprove: $A$ is a closed subset of $L^{2}$.

Let $\mathcal{F}=\left\{ f \in L^{2}\left(0,1\right): f([0,1]) \subset [0,1] \text{ a.e. }\right\}$. Prove or disprove: $\mathcal{F}$ is a closed subset of $L^{2}$.

Let $\left\{ f_{n}\right\} \subset A C[0,1]$, let $f_{n}(0)=0$ for all $n \in \mathbb {N}$, and suppose that $\left\{ f_{n}^{\prime }\right\}$ is a Cauchy sequence in $L^{1}\left(0,1\right)$. Prove that there exists $f \in A C[0,1]$ such that $\left\{ f_{n}\right\}$ converges to $f$ uniformly on $[0,1]$.

Let $A C[0,1]$ denote the normed space of absolutely continuous functions on $[0,1]$ with the norm $\left\| f\right\| :=\left|f(0)\right|+\int_{0}^{1}\left|f^{\prime }(t)\right| d t$. Prove that $A C[0,1]$ a Banach space with this norm.

Let $B V[0,1]$ denote the normed space of functions of bounded variation on $[0,1]$ with the norm $\left\| f\right\| :=\left|f(0)\right|+V_{0}^{1}(f)$. Prove that $B V[0,1]$ is a Banach space with this norm.

Let Lip $[0,1]$ denote the vector space of Lipschitz functions on $[0,1]$ with the norm

Prove that $\operatorname {Lip}[0,1]$ is a Banach space with this norm.

Prove that $\ell^{p}$ is a Banach space.

Prove that $\ell^{\infty }$ is a Banach space.

Let $1 \leq p<q \leq \infty$.

Let $1 \leq p \leq \infty$ and let $\left\{ f_{n}\right\}$ be a sequence in $L^{p}\left(0,1\right)$ such that

Let $1 \leq p \leq \infty$, let $g \in L^{p}\left(0,1\right)$, and let $\mathcal{F}$ be the subset of $L^{p}\left(0,1\right)$ consisting of functions $f$ that are not greater than $g$ a.e. Prove that $\mathcal{F}$ is a closed, nowhere dense subset of $L^{p}\left(0,1\right)$. Conclude that the set of all nonnegative functions in $L^{p}\left(0,1\right)$ is a closed, nowhere dense subset of $L^{p}\left(0,1\right)$.

Give an example of a sequence that converges uniformly but not in the norm of $L^{p}\left(X, \mu \right), 1 \leq p<\infty$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p<\infty$, and let $f$ and $\left\{ f_{n}\right\}$ be functions in $L^{p}\left(X, \mu \right)$ such that $\lim_{n}\left\| f_{n}\right\|_{p}=\left\| f\right\|_{p}$ and $\lim_{n} f_{n}=f$ in measure. Prove that $\left\{ f_{n}\right\}$ converges to $f$ in the norm of $L^{p}\left(X, \mu \right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space, let $1 \leq r<p<\infty$, and let $f$ and $\left\{ f_{n}\right\}$ be measurable functions such that $\sup_{n}\left\| f_{n}\right\|_{p}<\infty$ and $\lim_{n} f_{n}=f$ a.e. Prove that $\left\{ f_{n}\right\}$ converges to $f$ in the norm of $L^{r}\left(X, \mu \right)$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space, let $\left\{ E_{n}\right\} \subset \mathfrak {M}$ such that $\mu \left(E_{n}\right)< \infty$ for all $n \in \mathbb {N}$, and suppose that $\left\{ \chi_{E_{n}}\right\}$ converges to a function $f$ in the norm of $L^{1}\left(X, \mu \right)$. Prove that $f$ is equal a.e. to the characteristic function of a measurable set.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $0<p<1$, and for $f, g \in L^{p}\left(X, \mu \right)$ let

Prove that $d$ is a metric with respect to which $L^{p}\left(X, \mu \right)$ is complete.

By Hölder's Inequality, we have $L^{2}\left(a, b\right) \subset L^{1}\left(a, b\right)$. Show that the set $L^{2}\left(a, b\right)$, considered as a subset of a complete metric space $L^{1}\left(a, b\right)$, is of the first category.

Prove that $\left\| f\right\| =\left\| f\right\|_{1}+\left\| f^{\prime }\right\|_{1}$ defines a norm on $A C[0,1]$ and that it is a Banach space with respect to the norm.

Let $V$ be a closed linear subspace of $L^{1}\left(0,1\right)$. Suppose that for each $f \in V$ there exists a $p>1$ such that $f \in L^{p}\left(0,1\right)$. Show that there exists a $q>1$ such that $V \subset L^{q}\left(0,1\right)$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space, let $E$ be a nonempty measurable subset of $X$, and let $V$ consist of those functions in $L^{p}\left(X, \mu \right)$ that vanish on $E$. Prove that $V$ is a closed subspace of $L^{p}\left(X, \mu \right)$.

Prove or disprove: There is a nonempty closed set in $L^{p}\left(0,1\right)$ that does not contain an element of smallest norm.

For $p \in [1,+\infty ]$, prove that the closed unit ball of $L^{p}\left(0,1\right)$ is not compact.

Let $X$ be a Banach space, let $1 \leq p<\infty$ and define $\ell^{p}(X)$ to be the space of all sequences $\left\{ x_{n}\right\} \subset X$ such that $\sum_{n}\left\| x_{n}\right\|^{p}<\infty$. Prove that $\left\| \left\{ x_{n}\right\} \right\| :=\left(\sum_{n}\left\| x_{n}\right\|^{p}\right)^{1 / p}$ defines a norm on $\ell^{p}(X)$ and that it makes $\ell^{p}(X)$ a Banach space.