Normed Linear 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.