Banach Spaces

Prove that $\ell^{1}$ is a vector space and that $\left\| a\right\|_{1}=\sum_{k=1}^{\infty }\left|a_{k}\right|$ defines a norm on $\ell^{1}$.

Prove that $\ell^{\infty }$ is a vector space and that $\left\| a\right\|_{\infty }:=\sup_{n \in \mathbb {N}}\left|a_{n}\right|$ defines a norm on $\ell^{\infty }$.

Prove that $C[a, b]$ is a vector space and that $\left\| f\right\|_{\text{max }}=\max \left\{ \left|f(x)\right| : x \in [a, b]\right\}$ defines a norm on $C[a, b]$.

Let $A C[0,1]$ denote the vector space of absolutely continuous functions on $[0,1]$. (See Exercise 9.4.2.) Prove that $\left\| f\right\| :=\left|f(0)\right|+\int_{0}^{1}\left|f^{\prime }(t)\right| d t$ defines a norm on $A C[0,1]$.

Let $B V[0,1]$ denote the vector space of functions of bounded variation on $[0,1]$. (See Exercise 9.3.15.) Prove that $\left\| f\right\| :=\left|f(0)\right|+V_{0}^{1}(f)$ defines a norm on $B V[0,1]$.

Let Lip $[0,1]$ denote the vector space of Lipschitz functions on $[0,1]$. For $f \in \operatorname {Lip}[0,1]$ define

Prove that $\left\| \cdot \right\|$ is a norm.

Let $X$ be a finite-dimensional vector space with a basis $\left\{ x_{k}\right\}_{k=1}^{n}$. Given $x=\sum_{k=1}^{n} c_{k} x_{k}$ define

Prove that $\left\| \cdot \right\|_{\infty }$ is a norm on $X$.

Prove that $c$ is a vector space and that $\left\| \left(x_{1}, x_{2}, x_{3}, \ldots \right)\right\| =\sup \left\{ \left|x_{n}\right|: n \in \mathbb {N}\right\}$ defines a norm on $c$.

Let $X$ be a vector space and suppose that $\left\| \cdot \right\|_{\alpha }$ and $\left\| \cdot \right\|_{\beta }$ are both norms on $X$. Prove that so is $\left\| \cdot \right\| :=\left\| \cdot \right\|_{\alpha }+\left\| \cdot \right\|_{\beta }$.

Let $X$ be a normed linear space. Prove that $X$ is a Banach space if and only if for any $\left\{ x_{n}\right\} \subset X$ the convergence of $\sum_{n=1}^{\infty }\left\| x_{n}\right\|$ implies the convergence of $\sum_{n=1}^{\infty } x_{n}$.

Let $X$ be a normed linear space. Prove that $X$ is a Banach space if and only if the set $\left\{ x \in X:\left\| x\right\| =1\right\}$ is a complete metric space with respect to distance $d(x, y):=\left\| x-y\right\|$.

Prove that a normed linear space is complete if and only if every decreasing sequence of closed balls has a nonempty intersection.

Let $X$ be a normed space and $V$ a closed subspace of $X$. Prove that:

Prove that $C^{1}[a, b]$ is a vector space and that the formula $\left\| f\right\|_{\text{max }}= \max_{x \in [a, b]}\left|f(x)\right|$ defines a norm on $C^{1}[a, b]$, but that it is not a Banach space.

Let $c_{0}$ denote the subspace of $c$ consisting of sequences converging to 0. Prove that $c_{0}$ is a Banach space with respect to the norm (10.2).

Formula (10.2), from Example 10.1.9: $c$ is the normed space of convergent sequences of real numbers with norm $\left\| (x_{1}, x_{2}, x_{3}, \ldots )\right\| =\sup \left\{ \left|x_{n}\right|: n \in \mathbb {N}\right\}$.

Let $B(X)$ be the set of all bounded, real-valued functions on a set $X$. Prove that $B(X)$ is a Banach space with respect to the norm $\left\| f\right\| = \sup \left\{ \left|f(x)\right|: x \in X\right\}$.

For $f \in C[a, b]$ define

In Exercises 10.1.18-10.1.21, prove that the set of $x=\left\{ x_{n}\right\}$ for which $\left\| x\right\| <\infty$ is a Banach subspace of $c$ (the space of convergent sequences), where:

$\left\| x\right\| :=\left|x_{1}\right|+\sum_{n=1}^{\infty }\left|x_{n}-x_{n+1}\right|$.

In Exercises 10.1.18-10.1.21, prove that the set of $x=\left\{ x_{n}\right\}$ for which $\left\| x\right\| <\infty$ is a Banach subspace of $c$ (the space of convergent sequences), where:

$\left\| x\right\| :=\sum_{n=1}^{\infty }\left|x_{n}-x_{n+1}\right|$.

In Exercises 10.1.18-10.1.21, prove that the set of $x=\left\{ x_{n}\right\}$ for which $\left\| x\right\| <\infty$ is a Banach subspace of $c$ (the space of convergent sequences), where:

$\left\| x\right\| :=\sup_{n}\left|\sum_{k=1}^{n} x_{k}\right|$.

In Exercises 10.1.18-10.1.21, prove that the set of $x=\left\{ x_{n}\right\}$ for which $\left\| x\right\| <\infty$ is a Banach subspace of $c$ (the space of convergent sequences), where:

$\left\| x\right\| :=\sup_{n}\left|\sum_{k=1}^{n} x_{k}\right|$ and the space consists of $x \in c$ for which $\sum_{n=1}^{\infty } x_{n}$ converges.

Let $(X, \mathfrak {M})$ be a measurable space and let $M(\mathfrak {M})$ be the vector space of all finite signed measures on $\mathfrak {M}$. Prove that:

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

Let $X$ be a vector space of polynomials on $\mathbb {R}$ and, for a polynomial $p(x)=\sum_{k=0}^{\operatorname {deg} p} c_{k} x^{k}$, define

Prove that $\left\| \right\|$ is a norm and that $X$ is not complete with respect to this norm.

Let $X, Y$ be Banach spaces and for $(x, y) \in X \times Y$ define $\left\| (x, y)\right\| = \left\| x\right\|_{X}+\left\| y\right\|_{Y}$. Prove that $\left\| \right\|$ is a norm on $X \times Y$ which makes $X \times Y$ a Banach space.

Let $C_{b}(\mathbb {R})$ be the set of all bounded functions in $C(\mathbb {R})$ Prove that $C_{b}(\mathbb {R})$ is a Banach space.

Let $C_{0}(\mathbb {R}) \subset C(\mathbb {R})$ be defined by $f \in C_{0}(\mathbb {R})$ if and only if for every $\varepsilon >0$ the set $\left\{ x \in \mathbb {R}:\left|f(x)\right| \geq \varepsilon \right\}$ is compact. Prove that $C_{0}(\mathbb {R})$ is a Banach space.

Let $X$ be a subset of $\ell^{1}$ consisting of sequences $\left\{ a_{n}\right\}$ for which $\sum n\left|a_{n}\right|<\infty$. Prove that $X$ is a subspace of $\ell^{1}$ that is not closed.

Let $X$ be a normed space and $V, W$ two subspaces of $X$. Prove that $V+W$ is a subspace of $X$ and that $V+W=\vee (V \cup W)$.

Let $X$ be a normed space and $\left\{ V_{i}\right\}_{i \in I}$ a collection of subspaces of $X$. Prove that $\cap_{i \in I} V_{i}$ is a subspace of $X$.

Prove that every convergent sequence in a normed space $X$ is a Cauchy sequence.

Let $X$ be a normed space and $f: X \rightarrow \mathbb {R}$ a uniformly continuous function. Prove that there are $A, B>0$ such that for any $x \in X$,

Prove that the set consisting of the sequence $e=(1,1,1, \ldots )$ and the sequences $e_{n}=(0,0, \ldots , 0,1,0,0, \ldots )$ (with 1 in the $n$th position) has a dense span in $c$. (See Example 10.1.9.)

Example 10.1.9: Let $c$ denote the normed space of convergent sequences of real numbers with norm $\left\| (x_{1}, x_{2}, x_{3}, \ldots )\right\| =\sup \left\{ \left|x_{n}\right|: n \in \mathbb {N}\right\}$. This is a Banach space.

Let $e_{n}=(0,0, \ldots , 0,1,0,0, \ldots )$ (with 1 in the $n$th position). Prove that the set $\left\{ e_{n}\right\}_{n=1}^{\infty }$ has a dense span in $c_{0}$. (See Exercise 10.1.15.)

Let $X$ be a normed linear space. Prove that the function $F: X \rightarrow \mathbb {R}$ defined by $F(x)=\left\| x\right\|$ is continuous.

Let $X$ be a normed space. Prove that the closed unit ball is a closed subset of $X$.

Let $X$ be a normed space. Prove that the closed unit ball is the closure of the open unit ball.

Let $X$ be a normed space. Prove that the unit ball of $X$ has a unique center and radius.

Let $X$ be a Banach space and define $c_{0}(X)$ to be the space of all sequences $\left\{ x_{n}\right\} \subset X$ such that $\lim_{n}\left\| x_{n}\right\| =0$. Prove that $\left\| \left\{ x_{n}\right\} \right\| :=\sup_{n}\left\| x_{n}\right\|$ defines a norm on $c_{0}(X)$ and that it makes $c_{0}(X)$ a Banach space.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p<\infty$, and for a function $f: X \rightarrow \mathbb {R}^{*}$ let $[f]$ denote the equivalence class consisting of all functions that are equal to $f$ a.e. Prove that operations defined by formulas $[f]+[g]=[f+g]$ and $[\alpha f]=\alpha [f]$, where $\alpha \in \mathbb {R}$, are well defined and the same is true of $\left\| [f]\right\|_{p}=\left\| f\right\|_{p}$.

Prove Proposition 10.2.5.

Proposition 10.2.5: $L^{\infty }\left(X, \mu \right)$ is a vector space.

Prove Proposition 10.2.6.

Proposition 10.2.6: $L^{1}\left(X, \mu \right)$ is a normed vector space with the norm defined by $\left\| f\right\|_{1} := \int_{X} \left|f\right| \, d\mu$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p<\infty$, and let $\mathcal{L}^{p}$ be the set of all functions such that $\int_{X}\left|f\right|^{p} d \mu <\infty$. Prove that $d(f, g):=\left(\int_{X}\left|f-g\right|^{p}\right)^{1 / p}$ does not define a distance on $\mathcal{L}^{p}$. Formulate and prove a similar result for the space of all bounded functions.

Verify the assertion made in Example 10.2.16.

Example 10.2.16 (the inclusions $L^{p_{2}}\left(E\right) \subset L^{p_{1}}\left(E\right)$ are all proper): Let $E=(0,1]$, $-1/p_{1} < \alpha < -1/p_{2}$, and $f(x)=x^{\alpha }$. Then $f \in L^{p_{1}}\left(E\right) \setminus L^{p_{2}}\left(E\right)$.

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

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $f \in L^{\infty }\left(X, \mu \right)$. Prove that

Show that if $f(x)=\ln (1 / x)$ for $x \in (0,1]$, then $f$ belongs to $L^{p}(0,1]$ for all $1 \leq p<\infty$ but not to $L^{\infty }(0,1]$.

Let $f$ be an absolutely continuous function on $[0,1]$, such that $f(0)=0$ and

Prove that

exists and calculate the value of this limit.

Let $(X, \mathfrak {M}, \mu )$ be a measure space for which $\mu (X)=1$ and let $f$ be a measurable function on $X$. Prove that the function $F(p):=\left\| f\right\|_{p}$ is an increasing function on $[1, \infty )$. Under what circumstances is $F$ strictly increasing?

Prove or disprove: there exists a function $f$ that is bounded and continuous on $(0,+\infty )$ and $\lim_{x \rightarrow \infty } f(x)=0$ but $f \notin L^{p}\left(0, \infty \right)$ for any $p>0$.

Prove that the equality in Young's Inequality holds if and only if $b=a^{p / q}$.

Prove that the equality holds in Hölder's Inequality, i.e., $\int_{X}\left|f g\right|= \left\| f\right\|_{p}\left\| g\right\|_{q}$, if and only if $\left\| g\right\|_{q}^{q}\left|f\right|^{p}=\left\| f\right\|_{p}^{p}\left|g\right|^{q}$ a.e.

Prove that, for $1<p<\infty$, the equality in Minkowski's Inequality (10.7) holds if and only if there are nonnegative numbers $a, b$, not both zero, such that $a f=b g$ a.e.

Theorem 10.2.12 (Minkowski's Inequality): Let $(X, \mathfrak {M}, \mu )$ be a measure space and $1 \leq p \leq \infty$. If $f, g \in L^{p}\left(X, \mu \right)$ then so is $f+g$ and

Prove that, for $p=1$, the equality in Minkowski's Inequality (10.7) holds if and only if $f g \geq 0$ a.e.

Theorem 10.2.12 (Minkowski's Inequality): Let $(X, \mathfrak {M}, \mu )$ be a measure space and $1 \leq p \leq \infty$. If $f, g \in L^{p}\left(X, \mu \right)$ then so is $f+g$ and

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p<q<r<\infty$, and let $f \in L^{q}\left(X, \mu \right)$. Prove that for every $t>0$ there are functions $g \in L^{p}\left(X, \mu \right)$ and $h \in L^{r}\left(X, \mu \right)$ such that $f=g+h$ and

Prove Proposition 10.2.15.

Proposition 10.2.15: Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and $1 \leq p_{1} < p_{2} \leq \infty$. Then $L^{p_{2}}\left(X, \mu \right) \subset L^{p_{1}}\left(X, \mu \right)$. Furthermore, $\left\| f\right\|_{p_{1}} \leq C \left\| f\right\|_{p_{2}}$ for all $f \in L^{p_{2}}\left(X, \mu \right)$, where $C = \left(\mu \left(X\right)\right)^{(p_{2}-p_{1})/(p_{1} p_{2})}$ if $p_{2} < \infty$ and $C = \left(\mu \left(X\right)\right)^{1/p_{1}}$ if $p_{2} = \infty$.

Give an example of a measure space and numbers $p_{1}>p_{2} \geq 1$ such that $L^{p_{1}}(X)$ and $L^{p_{2}}(X)$ are equal.

Verify the assertion made in Remark 10.2.17.

Remark 10.2.17: Let $E \subset \mathbb {R}$ be measurable and $1 \leq p_{1}<p_{2} \leq \infty$. If $m(E)=\infty$, no inclusion holds between $L^{p_{1}}\left(E\right)$ and $L^{p_{2}}\left(E\right)$. (By contrast, by Proposition 10.2.15, if $m(E)<\infty$ then $L^{p_{2}}\left(E\right) \subset L^{p_{1}}\left(E\right)$.)

Let $0<\alpha \leq \beta$ and $f(x)=1 /\left(x^{\alpha }+x^{\beta }\right)$. Determine for what values of $p$ the function $f$ is in $L^{p}[0,+\infty )$.

Let $p, q, r$ be positive numbers satisfying $1 / p+1 / q+1 / r=1$ and suppose that $f \in L^{p}\left(X, \mu \right), g \in L^{q}\left(X, \mu \right), h \in L^{r}\left(X, \mu \right)$. Prove that the function $f g h$ is integrable and that $\left\| f g h\right\|_{1} \leq \left\| f\right\|_{p}\left\| g\right\|_{q}\left\| h\right\|_{r}$.

Let $1 \leq p \leq r \leq q \leq \infty$. Prove that

and that, if $f \in L^{p}\left(X, \mu \right) \cap L^{q}\left(X, \mu \right)$ then

and $q^{-1}$ is replaced by 0 if $q=\infty$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $f \in L^{\infty }\left(X, \mu \right)$. Prove that

Let $\left\{ f_{n}\right\}$ be a sequence of functions on $[0,1]$ such that $f_{n}$ is absolutely continuous for each $n \in \mathbb {N}$, $f_{n}(0)=0$ for each $n \in \mathbb {N}$, and the sequence $\left\{ f_{n}^{\prime }\right\}$ converges in $L^{1}\left(0,1\right)$.

Prove that:

For $p \in [1,+\infty ]$, find the values of the parameter $\lambda$ for which

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and for a measurable function $f$ let $E_{n}=\left\{ x \in X: n-1 \leq \left|f(x)\right|<n\right\}$. Prove that $f \in L^{p}\left(X, \mu \right), 1 \leq p<\infty$, if and only if

Is the finiteness of $\mu$ necessary?

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $0<p<1$. Prove that $\left\| \cdot \right\|_{p}$ is not a norm on $L^{p}\left(X, \mu \right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space such that $\mu (X)=1$, let $f, g \in L^{2}\left(X, \mu \right)$, and suppose that $\int_{X} f d \mu =0$. Prove that

Let $(X, \mathfrak {M}, \mu )$ be a measure space such that $\mu (X)=1$. For $1<p<\infty$ let

Prove that for each $0<\varepsilon <1$ there exists $\delta_{p}>0$ such that

for each $f \in \mathcal{A}_{p}$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $f$ be a nonnegative measurable function on $X$. Suppose that

Prove that $f=f^{2}$ a.e.

Let $\left\{ f_{n}\right\} \subset L^{p}\left(0,1\right),\left\{ g_{n}\right\} \subset L^{q}\left(0,1\right), 1 / p+1 / q=1$, and suppose that $\left\{ f_{n}\right\}$ converges to $f$ in $L^{p}\left(0,1\right)$ and that $\left\{ g_{n}\right\}$ converges to $g$ in $L^{q}\left(0,1\right)$. Prove or disprove:

Let $\left\{ f_{n}\right\}$ be a sequence of functions in $L^{\infty }\left(X\right)$ that converges a.e. to a function $f$, and suppose that $\left\| f_{n}\right\|_{\infty } \leq M$ for all $n \in \mathbb {N}$. Prove that $\left\| f\right\|_{\infty } \leq M$.

Let $E$ be a measurable subset of $[0,1]$ for which $m(E)>0$, and let $p<q$. Prove or disprove: There exists a function that belongs to $L^{p}\left(0,1\right)$ but not to $L^{q}\left(E\right)$.

Let $E$ be a measurable set and let $g$ be a function defined on $E$. Prove that the product $f g$ belongs to $L^{1}\left(E\right)$ for every function $f \in L^{1}\left(E\right)$ if and only if $g \in L^{\infty }\left(E\right)$.

Let $f$ be a function that belongs to both $L^{4}\left(\mathbb {R}\right)$ and $L^{8}\left(\mathbb {R}\right)$. Must $f$ belong to $L^{5}\left(\mathbb {R}\right)$ ? To $L^{2}\left(\mathbb {R}\right)$ ?

Prove or disprove: If $f \in L^{1}\left(0,+\infty \right)$ then there is an interval $(a, b)$ such that $f \in L^{2}\left(a, b\right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space for which $\mu (X)=1$ and let $p, q$ be two distinct numbers in $[1,+\infty ]$. Characterize those measurable, real-valued functions $f$ for which $\left\| f\right\|_{p}=\left\| f\right\|_{q}$.

Let $1 \leq p<q \leq \infty$. Prove that $\ell^{p} \subset \ell^{q}$. Is the inclusion strict?

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $1 \leq p<q \leq \infty$, and suppose that $f \in L^{r}\left(X, \mu \right)$ for all $r \in (p, q)$. Prove that

Let $(X, \mathfrak {M}, \nu )$ be a finite measure space, and let $\mu$ be a signed measure on $\mathfrak {M}$ such that $\left|\mu \right|$ is $\sigma$-finite and $\mu \ll \nu$. Prove that $d \mu / d \nu \in L^{p}\left(X, \nu \right)$ if and only if there is a constant $C>0$ such that

for any finite disjoint collection $\left\{ A_{k}\right\} \subset \mathfrak {M}$ satisfying $X=\cup_{k=1}^{n} A_{k}$ and $\nu \left(A_{k}\right)>0$, for $1 \leq k \leq n$. Determine the relationship between the smallest such constant $C$ and $\left\| d \mu / d \nu \right\|_{p}$.

Let $1 \leq p<\infty$ and let $f \in L^{p}\left(\mathbb {R}\right)$. Find $\lim_{\left|y\right| \rightarrow \infty } \int_{\mathbb {R}} \left|f(x+y)+f(x)\right|^{p} d x$.

Let $1 \leq p<\infty$. Prove or disprove: $\cap_{1 \leq r<p} L^{r}\left(0,1\right) \subset L^{p}\left(0,1\right)$.

Let $1 \leq p<\infty$. Prove or disprove: $\cup_{r>p} L^{r}\left(0,1\right) \supset L^{p}\left(0,1\right)$.

Let $1 \leq p<\infty$. Prove or disprove: $\cap_{r>p} L^{r}\left(\mathbb {R}\right) \subset L^{p}\left(\mathbb {R}\right)$.

Let $1 \leq p<\infty$. Prove or disprove: $\cup_{1 \leq r<p} L^{r}\left(\mathbb {R}\right) \supset L^{p}\left(\mathbb {R}\right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and $f \in L^{\infty }\left(X, \mu \right)$. The essential range of $f$ is the set $R_{f}$ of all $y \in \mathbb {R}$ for which the set $\left\{ x:\left|f(x)-y\right|<\varepsilon \right\}$ has positive measure for every $\varepsilon >0$. Prove that $R_{f}$ is compact and that $\left\| f\right\|_{\infty }=\sup \left\{ \left|y\right|: y \in R_{f}\right\}$.