Math Problems Vault

Problem 6A.1

Verify that each of the claimed metrics in Example 6.2 is indeed a metric.

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Remark.

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Example 6.2:

For a nonempty set $V$ , define $d$ on $V \times V$ by $d(f,g) = 1$ if $f \neq g$ and $d(f,g) = 0$ if $f = g$ . Then $d$ is a metric on $V$ .
Define $d$ on $\mathbb {R} \times \mathbb {R}$ by $d(x,y) = \left|x-y\right|$ . Then $d$ is a metric on $\mathbb {R}$ .
For $n \in \mathbb {N}$ , define $d$ on $\mathbb {R}^{n} \times \mathbb {R}^{n}$ by $d\left((x_1,\ldots ,x_n),(y_1,\ldots ,y_n)\right) = \max \left\{ \left|x_1-y_1\right|,\ldots ,\left|x_n-y_n\right|\right\}$ . Then $d$ is a metric on $\mathbb {R}^{n}$ .
Define $d$ on $C([0,1]) \times C([0,1])$ by $d(f,g) = \sup \left\{ \left|f(t)-g(t)\right| : t \in [0,1]\right\}$ , where $C([0,1])$ is the set of continuous real-valued functions on $[0,1]$ . Then $d$ is a metric on $C([0,1])$ .
Define $d$ on $\ell^{1} \times \ell^{1}$ by $d\left((a_1,a_2,\ldots ),(b_1,b_2,\ldots )\right) = \sum_{k=1}^{\infty }\left|a_k-b_k\right|$ , where $\ell^{1}$ is the set of sequences $(a_1,a_2,\ldots )$ of real numbers such that $\sum_{k=1}^{\infty }\left|a_k\right| < \infty$ . Then $d$ is a metric on $\ell^{1}$ .

Problem 6A.2

Prove that every finite subset of a metric space is closed.

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Problem 6A.3

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

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Problem 6A.4

(a)(b)

Suppose $V$ is a metric space.

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(a)

Prove that the union of each collection of open subsets of $V$ is an open subset of $V$ .

(b)

Prove that the intersection of each finite collection of open subsets of $V$ is an open subset of $V$ .

Problem 6A.5

(a)(b)

Suppose $V$ is a metric space.

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(a)

Prove that the intersection of each collection of closed subsets of $V$ is a closed subset of $V$ .

(b)

Prove that the union of each finite collection of closed subsets of $V$ is a closed subset of $V$ .

Problem 6A.6

(a)(b)

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(a)

Prove that if $V$ is a metric space, $f \in V$ , and $r > 0$ , then $\overline{B(f, r)} \subseteq \overline{B}(f, r)$ .

(b)

Give an example of a metric space $V, f \in V$ , and $r > 0$ such that $\overline{B(f, r)} \neq \overline{B}(f, r)$ .

Problem 6A.7

Show that each sequence in a metric space has at most one limit.

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Problem 6A.8

Prove 6.9.

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Remark.

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6.9 (closure): Suppose $V$ is a metric space and $E \subseteq V$ . Then

$\overline{E} = \left\{ g \in V: \text{there exist } f_{1}, f_{2}, \ldots \text{ in } E \text{ such that } \lim_{k \rightarrow \infty } f_{k} = g\right\}$ ;
$\overline{E}$ is the intersection of all closed subsets of $V$ that contain $E$ ;
$\overline{E}$ is a closed subset of $V$ ;
$E$ is closed if and only if $E = \overline{E}$ ;
$E$ is closed if and only if $E$ contains the limit of every convergent sequence of elements of $E$ .

Problem 6A.9

Prove that each open subset of a metric space $V$ is the union of some sequence of closed subsets of $V$ .

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Problem 6A.10

Prove or give a counterexample: If $V$ is a metric space and $U, W$ are subsets of $V$ , then $\overline{U} \cup \overline{W} = \overline{U \cup W}$ .

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Problem 6A.11

Prove or give a counterexample: If $V$ is a metric space and $U, W$ are subsets of $V$ , then $\overline{U} \cap \overline{W} = \overline{U \cap W}$ .

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Problem 6A.12

(a)(b)(c)

Suppose $\left(U, d_{U}\right),\left(V, d_{V}\right)$ , and $\left(W, d_{W}\right)$ are metric spaces. Suppose also that $T: U \rightarrow V$ and $S: V \rightarrow W$ are continuous functions.

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(a)

Using the definition of continuity, show that $S \circ T: U \rightarrow W$ is continuous.

(b)

Using the equivalence of 6.11(a) and 6.11(b), show that $S \circ T: U \rightarrow W$ is continuous.

(c)

Using the equivalence of 6.11(a) and 6.11(c), show that $S \circ T: U \rightarrow W$ is continuous.

Remark.

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6.11 (equivalent conditions for continuity): Suppose $V$ and $W$ are metric spaces and $T: V \rightarrow W$ is a function. Then the following are equivalent:

$T$ is continuous.
$\lim_{k \rightarrow \infty } f_{k} = f$ in $V$ implies $\lim_{k \rightarrow \infty } T(f_{k}) = T(f)$ in $W$ .
$T^{-1}(G)$ is an open subset of $V$ for every open set $G \subseteq W$ .
$T^{-1}(F)$ is a closed subset of $V$ for every closed set $F \subseteq W$ .

Problem 6A.13

Prove the parts of 6.11 that were not proved in the text.

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Remark.

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6.11 (equivalent conditions for continuity): Suppose $V$ and $W$ are metric spaces and $T: V \rightarrow W$ is a function. Then the following are equivalent:

$T$ is continuous.
$\lim_{k \rightarrow \infty } f_{k} = f$ in $V$ implies $\lim_{k \rightarrow \infty } T(f_{k}) = T(f)$ in $W$ .
$T^{-1}(G)$ is an open subset of $V$ for every open set $G \subseteq W$ .
$T^{-1}(F)$ is a closed subset of $V$ for every closed set $F \subseteq W$ . The text's own proof shows that (b) implies (d), and that (c) and (d) are equivalent.

Problem 6A.14

Suppose a Cauchy sequence in a metric space has a convergent subsequence. Prove that the Cauchy sequence converges.

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Problem 6A.15

Verify that all five of the metric spaces in Example 6.2 are complete metric spaces.

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Remark.

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Example 6.2:

For a nonempty set $V$ , define $d$ on $V \times V$ by $d(f,g) = 1$ if $f \neq g$ and $d(f,g) = 0$ if $f = g$ . Then $d$ is a metric on $V$ .
Define $d$ on $\mathbb {R} \times \mathbb {R}$ by $d(x,y) = \left|x-y\right|$ . Then $d$ is a metric on $\mathbb {R}$ .
For $n \in \mathbb {N}$ , define $d$ on $\mathbb {R}^{n} \times \mathbb {R}^{n}$ by $d\left((x_1,\ldots ,x_n),(y_1,\ldots ,y_n)\right) = \max \left\{ \left|x_1-y_1\right|,\ldots ,\left|x_n-y_n\right|\right\}$ . Then $d$ is a metric on $\mathbb {R}^{n}$ .
Define $d$ on $C([0,1]) \times C([0,1])$ by $d(f,g) = \sup \left\{ \left|f(t)-g(t)\right| : t \in [0,1]\right\}$ , where $C([0,1])$ is the set of continuous real-valued functions on $[0,1]$ . Then $d$ is a metric on $C([0,1])$ .
Define $d$ on $\ell^{1} \times \ell^{1}$ by $d\left((a_1,a_2,\ldots ),(b_1,b_2,\ldots )\right) = \sum_{k=1}^{\infty }\left|a_k-b_k\right|$ , where $\ell^{1}$ is the set of sequences $(a_1,a_2,\ldots )$ of real numbers such that $\sum_{k=1}^{\infty }\left|a_k\right| < \infty$ . Then $d$ is a metric on $\ell^{1}$ .

Problem 6A.16

Suppose $(U, d)$ is a metric space. Let $W$ denote the set of all Cauchy sequences of elements of $U$ .

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(a)

For $\left(f_{1}, f_{2}, \ldots \right)$ and $\left(g_{1}, g_{2}, \ldots \right)$ in $W$ , define $\left(f_{1}, f_{2}, \ldots \right) \equiv \left(g_{1}, g_{2}, \ldots \right)$ to mean that

\lim _{k \rightarrow \infty } d\left(f_{k}, g_{k}\right) = 0

Show that $\equiv$ is an equivalence relation on $W$ .

(b)

Let $V$ denote the set of equivalence classes of elements of $W$ under the equivalence relation above. For $\left(f_{1}, f_{2}, \ldots \right) \in W$ , let $\left(f_{1}, f_{2}, \ldots \right)^{\wedge }$ denote the equivalence class of $\left(f_{1}, f_{2}, \ldots \right)$ . Define $d_{V}: V \times V \rightarrow [0, \infty )$ by

d_{V}\left(\left(f_{1}, f_{2}, \ldots \right)^{\wedge },\left(g_{1}, g_{2}, \ldots \right)^{\wedge }\right) = \lim _{k \rightarrow \infty } d\left(f_{k}, g_{k}\right)

Show that this definition of $d_{V}$ makes sense and that $d_{V}$ is a metric on $V$ .

(c)

Show that $\left(V, d_{V}\right)$ is a complete metric space.

(d)

Show that the map from $U$ to $V$ that takes $f \in U$ to $(f, f, f, \ldots )^{\wedge }$ preserves distances, meaning that

d(f, g) = d_{V}\left((f, f, f, \ldots )^{\wedge },(g, g, g, \ldots )^{\wedge }\right)

for all $f, g \in U$ .

(e)

Explain why (d) shows that every metric space is a subset of some complete metric space.