Metric Spaces

Prove that the set $\mathbb {R}$ is a metric space with

Prove that the set $C[a, b]$ is a metric space with

Prove that the set $C[a, b]$ is a metric space with

Prove Theorem 2.1.6.

Theorem 2.1.6: The Euclidean norm has the following properties: (a) for any $a \in \mathbb {R}^{n}$, $\left\| a\right\| \geq 0$; (b) $\left\| a\right\| =0$ if and only if $a=0$; (c) if $\alpha \in \mathbb {R}$ and $a \in \mathbb {R}^{n}$, $\left\| \alpha a\right\| =\left|\alpha \right|\left\| a\right\|$; (d) if $a, b \in \mathbb {R}^{n}$, $\left\| a+b\right\| \leq \left\| a\right\| +\left\| b\right\|$.

Prove Theorem 2.1.7.

Theorem 2.1.7: The set $\mathbb {R}^{n}$ with the distance function $d(x,y)=\left\| y-x\right\|$ is a metric space.

Prove that $\mathbb {R}^{2}$ with the distance $d\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right):=\left|x_{1}-y_{1}\right|+\left|x_{2}-y_{2}\right|$ is a metric space.

Prove that $\mathbb {R}^{2}$ with the distance $d\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right):=\max \left\{ \left|x_{1}-y_{1}\right|,\left|x_{2}-y_{2}\right|\right\}$ is a metric space.

Prove that the set $\mathbb {R}$ is a metric space with

Prove that the set $C^{1}[a, b]$ of differentiable functions on $[a, b]$ with continuous derivatives, is a metric space with $d(f, g):=\max \left\{ \left|f(x)-g(x)\right|: a \leq x \leq b\right\} +\max \left\{ \left|f^{\prime }(x)-g^{\prime }(x)\right|: a \leq x \leq b\right\}$.

Prove that the set $C(-\infty , \infty )$ is a metric space with

Determine the values of $p$ for which $\left|x-y\right|^{p}$ is a metric on $\mathbb {R}$.

Let $M$ be any nonempty set and define $d: M \rightarrow \mathbb {R}$ by $d(x, y)=0$ if and only if $x=y$ and $d(x, y)=1$ otherwise. Prove that ( $M, d$ ) is a metric space

Let $c$ be the set of all bounded sequences and, for $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\} \in c$, let $d\left(\left\{ a_{n}\right\} ,\left\{ b_{n}\right\} \right)=\sup \left\{ \left|a_{n}-b_{n}\right|: n \in \mathbb {N}\right\}$. Prove that $(c, d)$ is a metric space.

Suppose that, for $1 \leq i \leq n$, ( $M_{i}, d_{i}$ ) are metric spaces. Let $M= M_{1} \times M_{2} \times \cdots \times M_{n}$ and for $x=\left(x_{1}, x_{2}, \ldots , x_{n}\right), y=\left(y_{1}, y_{2}, \ldots , y_{n}\right) \in M$ let $d(x, y)=\sum_{k=1}^{n} d_{k}\left(x_{k}, y_{k}\right)$. Prove that ( $M, d$ )is a metric space.

Describe open and closed balls in the metric space $\left(\mathbb {R}^{2}, d\right)$, where the distance $d$ is defined by $d\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right):=\left(\left|x_{1}-y_{1}\right|^{3}+\left|x_{2}-y_{2}\right|^{3}\right)^{1 / 3}$.

Describe open and closed balls in the metric space ( $C[0,1], d$ ), where $d(f, g):=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,1]\right\}$.

Let $u(x)=\sin x$ and $v(x)=0$ and let $C[0,2 \pi ]$ denote the metric space of continuous functions on $[0,2 \pi ]$ with the distance $d(f, g):=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,2 \pi ]\right\}$. Give an example of a function that belongs to both $B(u, 2 / 3)$ and $B(v, 2 / 3)$.

In the metric space $(\mathbb {R}, d)$ where

describe open balls $B(0, r)$ for:

Prove that $G:=\left\{ (x, y) \in \mathbb {R}^{2}: x^{2}+y^{2} \neq 1\right\}$ is an open set in $\mathbb {R}^{2}$.

Prove that an open ball in a metric space is an open set.

Prove that the following are equivalent for a set $A$ in a metric space:

Prove that in a metric space:

Prove that the interior of the interval $[a, b)$ is the open interval $(a, b)$.

Prove that the set $\mathbb {Q}$ has empty interior.

Let ( $M, d$ ) be a metric space. Prove that:

Let $A, B$ be subsets of $\mathbb {R}$. Prove that $\operatorname {Int}(A \times B)=\operatorname {Int} A \times \operatorname {Int} B$.

Let ( $M, d$ ) be a metric space. Prove that:

Prove that the set of all rational numbers between $\sqrt{2}$ and $\pi$ is a closed set in the metric space $\mathbb {Q}$ with the distance $d(x, y)=\left|x-y\right|$.

Prove that a closed ball in a metric space is a closed set.

Prove that in a metric space ( $M, d$ ) if $a \in M$ and $r>0$ then the set $\left\{ x: d(x, a)=r\right\}$ is closed.

Give an example of two closed sets $A, B$ such that $A \cap B=\varnothing$ but $\operatorname {dist}(A, B)=0$.

Prove that the set of irrational numbers is a $G_{\delta }$ set.

Prove that the closure of the interval $[a, b)$ is the closed interval $[a, b]$.

Prove that the closure of the set $\mathbb {Q}$ is $\mathbb {R}$.

Prove that in a metric space:

Prove that the following are equivalent for a set $A$ in a metric space:

Prove that the closure of a set $A$ equals the intersection of all closed sets that contain $A$.

Prove that in a metric space:

Prove or disprove: in a metric space ( $M, d$ ) the closure of an open ball $B(c, r)$ is the closed ball $B[c, r]$.

Prove that the boundary of the interval $[a, b)$ is the set $\left\{ a, b\right\}$.

Prove that the boundary of $\mathbb {Q}$ is the set $\mathbb {R}$.

Prove that the boundary of any set is closed.

Prove that a point $x$ in a metric space belongs to $\partial A$ if and only if every ball $B(x, r)$ has a nonempty intersection with both $A$ and $A^{C}$.

Prove that a set is closed if and only if it contains its boundary.

Prove that the interior of the boundary of a closed set is the empty set.

Give an example of a metric space in which the boundary of the open unit ball is a proper subspace of the unit sphere with the same center.

Prove that in a metric space every convergent sequence is a Cauchy sequence.

Prove that if a sequence $\left\{ x_{n}\right\}_{n=1}^{\infty }$ does not converge to $x$ then there is an open ball $B(x, r)$ that contains no members of the sequence.

Let $G$ be an open set in $\mathbb {R}^{n}$, let $x_{0} \in \mathbb {R}^{n}$, and let $\left\{ x_{n}\right\}$ be a sequence in $\mathbb {R}^{n}$ that converges to $x_{0}$. Prove that $G+x_{0} \subset \liminf \left(G+x_{n}\right)$.

Let $F$ be a closed set in $\mathbb {R}^{n}$, let $x_{0} \in \mathbb {R}^{n}$, and let $\left\{ x_{n}\right\}$ be a sequence in $\mathbb {R}^{n}$ that converges to $x_{0}$. Prove that $F+x_{0} \supset \limsup \left(F+x_{n}\right)$.

Let $A$ be a set in a metric space $M$ and let $a$ be a cluster point of $A$. Prove that there is a sequence $\left\{ a_{n}\right\} \subset A$ that converges to $a$.

Let ( $M, d$ ) be a metric space, and let $\lim x_{n}=x, \lim y_{n}=y$. Prove that $\lim d\left(x_{n}, y_{n}\right)=d(x, y)$.

In Exercises 2.2.7-2.2.9, $\left\{ a_{i j}\right\}_{i, j=1}^{\infty }$ is a sequence of nonnegative real numbers.

Prove that $\sum_{i=1}^{\infty } \sum_{j=1}^{\infty } a_{i j}=\sum_{j=1}^{\infty } \sum_{i=1}^{\infty } a_{i j}$.

In Exercises 2.2.7-2.2.9, $\left\{ a_{i j}\right\}_{i, j=1}^{\infty }$ is a sequence of nonnegative real numbers.

Prove that $\sup_{i=1}^{\infty } \sup_{j=1}^{\infty } a_{i j}=\sup_{j=1}^{\infty } \sup_{i=1}^{\infty } a_{i j}$.

In Exercises 2.2.7-2.2.9, $\left\{ a_{i j}\right\}_{i, j=1}^{\infty }$ is a sequence of nonnegative real numbers.

Prove that $\sup_{i=1}^{\infty } \sum_{j=1}^{\infty } a_{i j} \leq \sum_{j=1}^{\infty } \sup_{i=1}^{\infty } a_{i j}$. Give an example to show that the inequality may be strict.

Let $F \subset \mathbb {R}$ be a closed set that is contained in the disjoint union of intervals $(a, b)$ and $(c, d)$. Prove that $F \cap (a, b)$ is a closed set.

Let $A, B$ be closed subsets of $\mathbb {R}$. Prove that $A \times B$ is a closed subset of $\mathbb {R}^{2}$.

Prove that in a metric space every closed set is a $G_{\delta }$ set and that every open set is an $F_{\sigma }$ set.

Prove that the metric space $([0,1], d)$, where $d(x, y)=\left|x-y\right|$, is complete.

Prove that Euclidean space $\mathbb {R}^{n}$ is complete.

Prove that the metric space $(\mathbb {Q}, d)$, where $d(x, y)=\left|x-y\right|$, is not complete.

Prove that the set of all polynomials on $[0,1]$ with the metric $d(p, q):= \max \left\{ \left|p(x)-q(x)\right|: 0 \leq x \leq 1\right\}$ is not a complete metric space.

Prove that the set $C[0,1]$ with the metric

is not complete.

Prove that if a subspace of a metric space is complete then it is closed.

Give an example of a metric space ( $M, d$ ) and a closed set $A \subset M$ that is not complete.

Let $\left\{ x_{n}\right\}$ be a convergent sequence in a metric space. Prove that every subsequence of $\left\{ x_{n}\right\}$ converges to the same limit.

Let $\left\{ x_{n}\right\}$ be a sequence in a metric space and suppose that each of its three subsequences $\left\{ x_{2 n}\right\} ,\left\{ x_{2 n+1}\right\}$, and $\left\{ x_{3 n}\right\}$ is convergent. Prove that $\left\{ x_{n}\right\}$ is a convergent sequence.

Let $\left\{ x_{n}\right\}$ be a sequence in a metric space and suppose that each of its infinitely many subsequences $\left\{ x_{s n}\right\}$, where $s=2,3, \ldots$, is convergent. Prove or disprove: $\left\{ x_{n}\right\}$ is a convergent sequence.

Let $\left\{ x_{n}\right\}$ be a sequence in a metric space ( $M, d$ ), let $x \in M$, and suppose that each subsequence $\left\{ x_{n_{k}}\right\} \subset \left\{ x_{n}\right\}$ has a further subsequence $\left\{ y_{n}\right\}$ which converges to $x$. Prove that $\left\{ x_{n}\right\}$ converges to $x$.

Let ( $X, d$ ) be a metric space that is not complete and let $\left\{ x_{n}\right\}$ be a Cauchy sequence in $X$ that is not convergent. Prove that the set $E=\left\{ x_{n}\right\}$ is closed.

In Exercises 2.2.25-2.2.28, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in ( 0,1 ).

$x_{n}=1 / n$.

In Exercises 2.2.25-2.2.28, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in ( 0,1 ).

$1,0,1 / 2,0,1 / 3,0,1 / 4,0, \ldots$.

In Exercises 2.2.25-2.2.28, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in ( 0,1 ).

$x_{n}=e^{n} / n!$.

In Exercises 2.2.25-2.2.28, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in ( 0,1 ).

$1,1,1 / 4,1 / 4,1 / 9,1 / 9,1 / 16,1 / 16, \ldots$.

In Exercises 2.2.29-2.2.32, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in the metric space $C[0,1]$ (see Exercise 2.1.2).

$f_{n}(x)=x^{n}$

In Exercises 2.2.29-2.2.32, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in the metric space $C[0,1]$ (see Exercise 2.1.2).

$f_{n}(x)=x^{n} / n$

In Exercises 2.2.29-2.2.32, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in the metric space $C[0,1]$ (see Exercise 2.1.2).

$f_{n}(x)=x^{n} / \sqrt{n}$

In Exercises 2.2.29-2.2.32, determine which of the following sequences is (a) a Cauchy sequence, (b) a rapidly Cauchy sequence in the metric space $C[0,1]$ (see Exercise 2.1.2).

$f_{n}(x)=\sin n x$

Prove Theorem 2.2.23.

Theorem 2.2.23: A metric space is complete if and only if every rapidly Cauchy sequence converges. (A sequence $\left(x_{n}\right)_{n=1}^{\infty }$ in a metric space $(M,d)$ is rapidly Cauchy if there is a convergent series of positive numbers $\sum_{k=1}^{\infty } \varepsilon_{k}$ such that $d(x_{k+1}, x_{k}) \leq \varepsilon_{k}^{2}$ for all $k$.)

Prove that a Cauchy sequence in a metric space is bounded.

Prove that every totally bounded metric space is bounded.

Prove that every bounded subset of $\mathbb {R}^{n}$ is totally bounded.

Prove that the closure of a totally bounded set in a metric space is totally bounded.

Prove that a metric space ( $M, d$ ) is totally bounded if and only if every sequence in $M$ has a Cauchy subsequence.

Prove that the closure of a bounded set in a metric space is bounded.

Prove that a compact set in a metric space is closed.

Prove that a closed subset of a compact metric space is compact.

Prove that a metric space ( $M, d$ ) is compact if and only if every infinite subset of $M$ has a cluster point in $M$.

Prove that a subset of a complete metric space $M$ is compact if and only if it is totally bounded and closed.

In the metric space $(\mathbb {Q}, d)$, where $d(x, y)=\left|x-y\right|$, determine all subspaces that are:

Prove or disprove: the set $\mathbb {N}$ is a compact subset of ( $\mathbb {R}, d$ ), where $d(x, y)=\left|x-y\right|$.

Let ( $M, d$ ) be a complete metric space. Prove that a set $A \subset M$ is totally bounded if and only if $\bar{A}$ is compact.

Let ( $M, d$ ) be the metric space of all bounded sequences, with metric $d(a, b)=\sup \left\{ \left|a_{n}-b_{n}\right|: n \in \mathbb {N}\right\}$, where $a=\left\{ a_{n}\right\}$ and $b=\left\{ b_{n}\right\}$. (See Exercise 2.1.13.) Let $B$ be the closed ball of radius 1 centered at the zero sequence $(0,0, \ldots )$. Prove that $B$ is bounded and closed but not sequentially compact.

Let ( $M, d$ ) be a metric space and let $A \subset M$ be a set such that $A \cap K$ is compact for every compact set $K \subset M$. Prove that $A$ is closed in $M$.

Let ( $M, d$ ) be a metric space. Prove that the distance between a closed set and a compact set that are disjoint cannot be 0 . Give an example to show that this is false if both sets are only closed.

Let $K_{1}, K_{2}$ be disjoint compact subsets of a metric space ( $M, d$ ). Prove that there exist disjoint, open sets $G_{1}, G_{2}$ such that $K_{1} \subset G_{1}$ and $K_{2} \subset G_{2}$.

Prove that a metric space $M$ is compact if and only if every collection of closed subsets of $M$ with the finite intersection property (meaning that every finite subcollection has non-empty intersection) has a non-empty intersection.

Prove that a metric space $M$ is compact if and only if every decreasing sequence of nonempty closed sets in $M$ has nonempty intersection.

Prove that the union of a finite number of compact sets in a metric space is compact.

Let $M$ be a compact metric space and let $\left\{ G_{n}\right\}_{n=1}^{\infty }$ be an increasing sequence of open subsets of $M$ such that $M=\cup_{n=1}^{\infty } G_{n}$. Prove that $M=G_{k}$ for some $k \in \mathbb {N}$.

Let $\left\{ M_{i}\right\}_{i=1}^{n}$ be compact metric spaces. Prove that $M:=M_{1} \times M_{2} \times \cdots \times M_{n}$ is a compact metric space with respect to metric $d$ as in Exercise 2.1.14.

Determine if $C[0,1]$ is sequentially compact.

Determine if the metric space ( $M, d$ ) is sequentially compact, where $d(x, y)=0$ if $x=y$ and $d(x, y)=1$ if $x \neq y$.

Determine if the metric space ( $\mathbb {R}, d$ ) is sequentially compact, where

Determine if the metric space ( $C[0,1], d$ ) is sequentially compact, where

Determine if the metric space $(C(-\infty , \infty ), d)$ is sequentially compact, where

Prove Theorem 2.3.18.

Theorem 2.3.18: Let $(M,d)$ be a compact metric space and let $f: M \rightarrow \mathbb {R}$ be continuous. Then $f$ is bounded and it attains its extreme values on $M$.

Let ( $M, d$ ) be a metric space and let $f, g: M \rightarrow \mathbb {R}$ be continuous. Prove that $\max \left\{ f, g\right\}$ and $\min \left\{ f, g\right\}$ are continuous.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$, let $x_{0} \in \mathbb {R}$, and suppose that for every interval $(a, b)$ that contains $x_{0}$, the function $f$ is unbounded on ( $a, b$ ). Prove that $f$ is not continuous at $x_{0}$.

Let ( $M, d$ ) be a metric space, let $f: M \rightarrow \mathbb {R}$ be continuous, and let $E \subset M$. Prove that $\sup \left\{ \left|f(x)\right|: x \in E\right\} =\sup \left\{ \left|f(x)\right|: x \in \bar{E}\right\}$.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be a strictly increasing continuous function. Prove that $f^{-1}$ is continuous.

Let ( $M, d$ ) be a metric space, let $f, g: M \rightarrow \mathbb {R}$ be continuous, and let $\alpha \in \mathbb {R}$. Prove that $\alpha f, f+g$ and $f g$ are continuous functions. Moreover, if $g(x) \neq 0$ on $M$, prove that $f / g$ is continuous.

Let $f$ be a real-valued function on a metric space $X$. Show that the set of points at which $f$ is continuous is the intersection of a countable collection of open sets. Conclude that there is no real-valued function on $\mathbb {R}$ that is continuous at the rational numbers only.

Let ( $M, d$ ) be a metric space and $K$ a compact subset of $M$. Suppose that a function $f: K \rightarrow \mathbb {R}$ is continuous. Prove that $f(K)$ is a compact subset of $\mathbb {R}$.

Let ( $M, d$ ) be a metric space and $A \subset M$. Suppose that functions $f, g: M \rightarrow \mathbb {R}$ are continuous and $f(x)=g(x)$ when $x \in A$. Prove that $f(x)=g(x)$ for all $x \in \bar{A}$.

Let ( $M, d$ ) be a metric space, let $A \subset M$, and let $h(x)=d(x, A)= \inf \left\{ d(x, a): a \in A\right\}$. Prove that $d$ is a continuous function on $M$.

Let ( $M, d$ ) be a metric space and let $A$ and $B$ be disjoint, closed subsets of $M$. Prove that there exists a continuous, real-valued function $f$ on $M$ such that $0 \leq f(x) \leq 1$, for all $x \in M$, $f(x)=0$ on $A$, and $f(x)=1$ on $B$.

Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions that converge to $f$ uniformly on $[a, b]$ and let $\left\{ x_{n}\right\}$ be a sequence in $[a, b]$ that converges to $x$. Prove that $\left\{ f_{n}\left(x_{n}\right)\right\}$ converges to $f(x)$.

Prove Theorem 2.3.17.

Theorem 2.3.17: Let $(M,d)$ be a metric space. Every uniformly continuous map $f: M_{1} \rightarrow M_{2}$ is continuous. If, in addition, $M_{1}$ is compact then every continuous map $f: M_{1} \rightarrow M_{2}$ is uniformly continuous.

Determine whether the function $F$ in Example 2.3.12 is uniformly continuous on $C[0,1]$.

Example 2.3.12: Let $a \in [0,1]$. The function $F: C[0,1] \rightarrow \mathbb {R}$, defined by $F(f) = f(a)$, is continuous on the metric space $C[0,1]$.

Determine whether the function $F$ in Example 2.3.12 is continuous on ( $C[0,1], d$ ) with $d$ defined as in Exercise 2.1.3, by

Is it uniformly continuous?

Example 2.3.12: Let $a \in [0,1]$. The function $F: C[0,1] \rightarrow \mathbb {R}$, defined by $F(f)=f(a)$, is continuous on the metric space $C[0,1]$ with the metric $d(f, g)=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,1]\right\}$.

Let ( $M, d$ ) be a metric space and $\left\{ x_{n}\right\} \subset A \subset M$ a Cauchy sequence. Suppose that a function $f: M \rightarrow \mathbb {R}$ is uniformly continuous on $A$. Prove that $\left\{ f\left(x_{n}\right)\right\}$ is a Cauchy sequence in $\mathbb {R}$. What if $f$ is assumed to be only continuous?

Let ( $M, d$ ) be a metric space and let $f: M \rightarrow \mathbb {R}$. Prove that the following statements are equivalent:

Prove that the function $f=\chi_{(0,1]}$ is upper semicontinuous at every point except at $x=0$ and it is lower semicontinuous at every point except at $x=1$.

Let ( $M, d$ ) be a metric space. Suppose that, for every $\varepsilon >0$, there exists $\delta >0$, a metric space ( $W, \rho$ ), and a mapping $\Phi : M \rightarrow W$ so that $\Phi (M)$ is totally bounded, and whenever $x, y \in M$ are such that $\rho (\Phi (x), \Phi (y))<\delta$, then $d(x, y)<\varepsilon$. Prove that $M$ is totally bounded.