Math Problems Vault

Problem 3.18.1

Prove that for any three points $x, y, z$ of a metric space $X$ the inequality holds

\left|\rho (x ; z)-\rho (y ; z)\right| \leqslant \rho (x ; y)

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Problem 3.18.2

Prove that for any four points $x, y, u, v$ of a metric space $X$ the inequality holds

\left|\rho (x ; u)-\rho (y ; v)\right| \leqslant \rho (x ; y)+\rho (u ; v).

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Problem 3.18.3

Prove that the functions indicated below are metrics on the corresponding sets:

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(1)

$\rho (x ; y) = \left|x-y\right|$ on the set of all real numbers $R$ , $x, y \in R$

(2)

$\rho (z ; w) = \left|z-w\right|$ on the set of all complex numbers $C$ , $z, w \in C$ ;

(3)

$\rho (x ; y) = \sqrt{\sum_{i = 1}^{n}\left(x_{i}-y_{i}\right)^{2}}$ on the set of points of the $n$ -dimensional space $R^{n}, x = \left(x_{1} ; \ldots ; x_{n}\right), y = \left(y_{1} ; \ldots ; y_{n}\right) \in R^{n}$ ;

(4)

$\rho (x ; y) = \sup_{t \in E}\left|x(t)-y(t)\right|$ on the set $B_{C}(E)$ of all complex-valued functions bounded on an arbitrarily given set $E$ ;

(5)

$\rho (x ; y) = \int_{G}\left|x(t)-y(t)\right| d t$ on the set $C L_{1}(\bar{G})$ of all functions continuous on the closure $\bar{G}$ of a Jordan-measurable open set $G \subset R^{n}$ ;

(6)

$\rho (x ; y) = \int_{-\infty }^{+\infty }\left|x(t)-y(t)\right| d t$ on the set $C L_{1}(R)$ of all functions continuous and absolutely integrable on the real line $R$ ;

(7)

$\rho (x ; y) = \sqrt{\int_{G}[x(t)-y(t)]^{2} d t}$ on the set $C L_{2}(\bar{G})$ of all real functions continuous on the closure $\bar{G}$ of a measurable open set $G \subset R^{n}$ ;

(8)

$\rho (x ; y) = \sqrt{\int_{-\infty }^{+\infty }[x(t)-y(t)]^{2} d t}$ on the set $C L_{2}(R)$ of all real functions continuous on $R$ for which the integral $\int_{-\infty }^{+\infty }[f(t)]^{2} d t$ converges.

Problem 3.18.4

Prove that the set $l_{\infty }$ of all sequences $x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ of real numbers for which $\sup \left|x_{n}\right| < +\infty$ is a metric space with the metric $\rho (x ; y) = \sup \left|x_{n}-y_{n}\right|, x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right), y = \left(y_{1} ; \ldots ; y_{n} ; \ldots \right) \in l_{\infty }$ .

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Problem 3.18.5

Prove that the set $l_{p}, 1 \leqslant p < +\infty$ , of all sequences $x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ of real numbers for which $\sum_{n = 1}^{\infty }\left|x_{n}\right|^{p} < +\infty$ is a metric space with the metric

\rho (x ; y) = \left(\sum _{n = 1}^{\infty }\left|x_{n}-y_{n}\right|^{p}\right)^{1 / p}

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Problem 3.18.6

Will sequences $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ of complex numbers, with the metrics introduced in problems 4 and $5\left(x_{n}, y_{n} \in C, \quad n = 1, \ldots , 2, \ldots \right)$ , form metric spaces?

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Problem 3.18.7

Prove that the set $C L_{p}(\bar{G}), 1 \leqslant p < +\infty$ , of all functions continuous on the closure $\bar{G}$ of a Jordan-measurable open set $G \subset R^{n}$ , is a metric space with the metric

\rho (x ; y) = \left(\int _{G}|x(t)-y(t)|^{p} d t\right)^{1 / p}

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Problem 3.18.8

Is the function $\rho (x ; y) = \int_{a}^{b}\left|x(t)-y(t)\right| d t$ a metric on the set of all functions Riemann integrable on the segment $[a ; b]$ ?

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Problem 3.18.9

Let $(a ; b)$ be a finite or infinite interval: $-\infty \leqslant a < b \leqslant +\infty$ . Prove that the set $C L_{p}(a ; b)$ of all functions continuous on the interval $(a ; b)$ for which the integral $\int_{a}^{b}|x(t)|^{p} d t$ converges is a metric space with the metric

\rho (x ; y) = \left(\int _{a}^{b}|x(t)-y(t)|^{p} d t\right)^{1 / p}

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Problem 3.18.10

Is the set of all real numbers a metric space if the distance between $x$ and $y$ is understood to be $\sin^{2}(x-y)$ ?

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Problem 3.18.11

Is the set of points of a circle a metric space, if the distance between points is taken to be the length of the shortest arc joining the given points?

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Problem 3.18.12

Is the set of all functions continuously differentiable on the segment $[a ; b]$ a metric space, if

\rho (x ; y) = \max _{[a ; b]}\left|x^{\prime }(t)-y^{\prime }(t)\right|?

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Problem 3.18.13

Is a nonempty set $X$ a metric space if $\rho (x ; y) = 0$ for $x = y$ and $\rho (x ; y) = 1$ for $x \neq y$ ?

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Problem 3.18.14

Will the following function be a metric on the set of all numerical sequences $x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ ( $x_{n} \in R$ or $x_{n} \in C, n = 1,2, \ldots$ )

\rho (x ; y) = \sum _{n = 1}^{\infty } 2^{-n} \frac{\left|x_{n}-y_{n}\right|}{1+\left|x_{n}-y_{n}\right|}, \quad x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right), \quad y = \left(y_{1} ; \ldots ; y_{n} ; \ldots \right)?

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Problem 3.18.15

Is the family of all nonempty subsets of a metric space $X$ a metric space, if the "distance" between the sets $E_{1} \subset X$ and $E_{2} \subset X$ is defined by the equality

\rho \left(E_{1} ; E_{2}\right) = \inf _{x \in E_{1}, y \in E_{2}} \rho (x ; y)?

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Problem 3.18.16

Suppose that on the set of ordered pairs $(x ; y)$ of elements of a set $X$ a nonnegative function $\rho (x ; y)$ is defined, satisfying all the axioms of a metric except the first, which holds in the weakened form: for any element $x \in X$ the equality $\rho (x ; x) = 0$ holds. Call the elements $x$ and $y$ of the set $X$ equivalent if $\rho (x ; y) = 0$ . Let $X^{*} = \left\{ x^{*}\right\}$ be the set of all classes of equivalent elements of the set $X$ . Prove that the function $\rho^{*}\left(x^{*}, y^{*}\right) = \rho (x ; y)$ , where $x \in x^{*} \in X^{*}, y \in y^{*} \in X^{*}$ , does not depend on the choice of the elements $x$ and $y$ in the respective classes $x^{*}, y^{*}$ , and is a metric on the set $X^{*}$ .

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Problem 3.18.17

Prove that if $X$ and $Y$ are metric spaces with metrics $\rho_{X}$ and $\rho_{Y}$ respectively, then the function

\rho \left(\left(x_{1} ; y_{1}\right),\left(x_{2} ; y_{2}\right)\right) = \sqrt{\left[\rho _{X}\left(x_{1} ; x_{2}\right)\right]^{2}+\left[\rho _{Y}\left(y_{1} ; y_{2}\right)\right]^{2}}

is a metric on their product $X \times Y$ , called in this case the product of the metric spaces $X$ and $Y$ .

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Problem 3.18.18

Let $\rho (x ; y)$ be a metric on a set $X$ . Prove that the functions

\begin{gathered} \rho _{1}(x ; y) = \frac{\rho (x ; y)}{1+\rho (x ; y)} \\ \rho _{2}(x ; y) = \min \left\{ \rho (x, y) ; 1\right\} , \quad \rho _{3}(x, y) = \ln (1+\rho (x, y))\end{gathered}

are metrics on the set $X$ , equivalent to the metric $\rho$ .

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Problem 3.18.19

Give an example of a sequence of functions continuous on the segment $[a ; b]$ , converging in the space $C L_{p}[a ; b], 1 \leqslant p < +\infty$ (see problem 3.7), but not converging uniformly on the segment $[a ; b]$ . (In the case of a segment we shall write $C L_{p}[a ; b]$ instead of $C L_{p}([a ; b]), 1 \leqslant p < +\infty$ ; see problem 7.)

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Problem 3.18.20

Give an example of a sequence of functions belonging to the space $C L_{2}[0 ; 1]$ , converging in the space $C L_{1}[0 ; 1]$ , but not converging in the space $C L_{2}[a ; b]$ .

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Problem 3.18.21

Give an example of a sequence of continuous functions, converging at every point of the segment $[a ; b]$ , but not converging in the space $C L_{2}[a ; b]$ .

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Problem 3.18.22

Give an example of a sequence of functions, converging in the space $C L_{1}[0 ; 1]$ , but not converging at any point of the segment $[0 ; 1]$ .

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Problem 3.18.23

Will the metrics of the spaces $C L_{1}[a ; b]$ and $C L_{2}[a ; b]$ be equivalent on the set of functions continuous on the segment $[a ; b]$ ?

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Problem 3.18.24

Prove that for the equivalence of the metrics $\rho_{1}$ and $\rho_{2}$ on a set $X$ it suffices that there exist two constants $c_{1} > 0$ and $c_{2} > 0$ such that for all $x \in X$ and $y \in Y$ the inequality $c_{1} \rho_{1}(x ; y) \leqslant \rho_{2}(x ; y) \leqslant c_{2} \rho_{1}(x ; y)$ holds. Show that this condition is not necessary for the equivalence of the metrics $\rho_{1}$ and $\rho_{2}$ .

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Problem 3.18.25

Prove that a sequence in a metric space can have only one limit.

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Problem 3.18.26

Prove that the set of values of a convergent sequence of points of a metric space is a bounded set.

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Problem 3.18.27

Can a sequence of functions $x_{n}:[0 ; 1] \rightarrow R, n = 1,2, \ldots$ , converging in the space $C L_{2}[0 ; 1]$ , be unbounded? (See problem 3.7.)

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Problem 3.18.28

If $x^{(m)} = \left(x_{1}^{(m)} ; \ldots ; x_{n}^{(n)} ; \ldots \right) \in l_{\infty }, m = 1,2, \ldots$ (see problem 4 $)$ and for each $n = 1,2, \ldots$ there exists a finite limit $\lim_{m \rightarrow \infty } x_{n}^{(m)} = a_{n}$ , will the sequence $\left(a_{1} ; \ldots ; a_{n} ; \ldots \right)$ always belong to $l_{\infty }$ ? If $\lim_{m \rightarrow \infty } x_{n}^{(m)} = a_{n}$ and $a = \left(a_{1} ; \ldots ; a_{n} ; \ldots \right) \in l_{\infty }$ , will the assertion that $\lim_{m \rightarrow \infty } x^{(m)} = a$ in $l_{\infty }$ be true?

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Problem 3.18.29

Prove that for any two distinct points of a metric space there exist disjoint balls centered at these points.

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Problem 3.18.30

Prove that a set $E$ of a metric space is bounded if and only if for every point $x \in X$ there exists $\varepsilon > 0$ such that the ball centered at the point $x$ of radius $\varepsilon$ contains the set $E$ .

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Problem 3.18.31

Prove that if a set is bounded in the space $C[a ; b]$ of functions continuous on the segment $[a ; b]$ with the metric

\rho (x ; y) = \max _{[a ; b]}\left|x(t)-y(t)\right|, \quad x, y \in C[a ; b]

then it is also bounded in any space $C L_{p}[a ; b], 1 \leqslant p < +\infty$ (see problem 7).

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Problem 3.18.32

Is it true that if a set of functions continuous on the segment $[a ; b]$ is bounded in some space $C L_{p}[a ; b], 1 \leqslant p < +\infty$ , then it is also bounded in the space $C[a ; b]$ ? (See problem 31.)

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Problem 3.18.33

(1)(2)

Under what conditions on the sequence $a_{n} \in R, a_{n} > 0$ , $n = 1,2, \ldots$ , will the following sets be bounded in the space $l_{2}$ (see problem 24.5):

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(1)

the parallelepiped $\left\{ x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in l_{2}|| x_{n} \mid < a_{n}, n = 1,2, \ldots \right\}$ ;

(2)

the ellipsoid $\left\{ x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in l_{2} \mid \sum_{n = 1}^{\infty } \frac{x_{n}^{2}}{a_{n}^{2}} < 1\right\}$ ?

Problem 3.18.34

Can a ball of radius 4 be a proper subset of a ball of radius 3?

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Problem 3.18.35

Prove that the $\varepsilon$ -neighborhood of a point of a metric space is an open set.

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Problem 3.18.36

Prove that the set of interior points of any set of a metric space is an open set.

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Problem 3.18.37

Prove that the set of interior points of any set of a metric space is the union of all open sets contained in the given set.

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Problem 3.18.38

Give an example of a set open in a subspace of a metric space that is not open in the space itself.

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Problem 3.18.39

Give an example of a set closed in a subspace of a metric space that is not closed in the space itself.

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Problem 3.18.40

Prove the stated assertion. A subset $E$ of a subspace $Y$ of a metric space $X$ is open (closed) if and only if it is the intersection of a set open (closed) in $X$ with the subspace $Y$ .

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Problem 3.18.41

Prove the stated assertion. If $G$ is an open set and $F$ a closed set of a metric space $X$ , then $G \backslash F$ is a set open in $X$ .

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Problem 3.18.42

Prove the stated assertion. The intersection of a finite collection and the union of any collection of open sets are open sets. Give an example of an infinite collection of open sets whose intersection is not an open set.

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Problem 3.18.43

Prove the stated assertion. The union of a finite collection and the intersection of any collection of closed sets are closed sets.

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Problem 3.18.44

Prove the stated assertion. For a point $x$ of a metric space $X$ to be a point of closure of a set $E \subset E$ , it is necessary and sufficient that there exist a sequence of points $x_{n} \in E, n = 1,2, \ldots$ , such that $\lim_{n \rightarrow \infty } x_{n} = x$ .

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Problem 3.18.45

Prove the stated assertion. For a point $x$ of a metric space $X$ to be a limit point of a set $E \subset X$ , it is necessary and sufficient that every neighborhood of the point $x$ contain infinitely many points of the set $E$ .

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Problem 3.18.46

Prove the stated assertion. A set of a metric space is closed if and only if it contains the set of all its limit points.

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Problem 3.18.47

Prove the stated assertion. The closure of a set in a metric space is a closed set.

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Problem 3.18.48

Prove the stated assertion. A closed ball of a metric space is a closed set.

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Problem 3.18.49

Can a closed ball fail to be the closure of the open ball with the same center and radius?

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Problem 3.18.50

(1)(2)

Are the following statements true:

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(1)

the set of interior points of the intersection of two sets is the intersection of the sets of their interior points;

(2)

the set of interior points of the intersection of any collection of sets is the intersection of the sets of their interior points?

If not, does the inclusion hold in either direction?

Problem 3.18.51

(1)(2)

Are the following statements true:

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(1)

the set of interior points of the union of two sets is the union of the sets of their interior points;

(2)

the set of interior points of the union of any collection of sets is the union of the sets of their interior points?

If not, does the inclusion hold in either direction?

Problem 3.18.52

(1)(2)

Are the following statements true:

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(1)

the closure of the union of two sets is the union of their closures;

(2)

the closure of the union of any collection of sets is the union of the closures of these sets?

If not, does the inclusion hold in either direction?

Problem 3.18.53

(1)(2)

Are the following statements true:

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(1)

the closure of the intersection of two sets is the intersection of their closures;

(2)

the closure of the intersection of any collection of sets is the intersection of the closures of these sets?

If not, does the inclusion hold in either direction?

Problem 3.18.54

Prove the stated assertion. The closure of a set is the intersection of all closed sets containing the given set.

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Problem 3.18.55

Prove the stated assertion. The boundary of a set is a closed set.

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Problem 3.18.56

Prove the stated assertion. For any set, $\bar{E} = E \cup \partial E$ .

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Problem 3.18.57

Prove the stated assertion. A nonempty subset of a metric space is open if and only if it does not intersect its boundary.

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Problem 3.18.58

Prove the stated assertion. A set is closed if and only if it contains all its boundary points.

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Problem 3.18.59

Prove the stated assertion. The boundaries of the union, intersection, and difference of two sets are contained in the union of the boundaries of these sets.

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Problem 3.18.60

Construct an example of an infinite collection of sets whose union's boundary is not contained in the union of their boundaries.

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Problem 3.18.61

Prove the stated assertion. The set of limit points of a set is closed.

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Problem 3.18.62

Prove the stated assertion. For any set, $\operatorname {diam} E = \operatorname {diam} \bar{E}$ .

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Problem 3.18.63

Prove the stated assertion. If $x(t)$ is a function continuous on a metric space $X$ , then for any number $a \in R$ the sets $\left\{ t \in X \mid x(t) \leqslant a\right\}$ and $\left\{ t \in X \mid x(t) \geqslant a\right\}$ are closed, and the sets $\left\{ t \in X \mid x(t) < a\right\}$ and $\left\{ t \in X \mid x(t) > a\right\}$ are open.

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Problem 3.18.64

Prove the stated assertion. If $x(t)$ is a function continuous on a subset $E$ of a metric space $X$ , then for any numbers $a \in R$ the sets $\left\{ t \in E \mid x(t) \leqslant a\right\}$ and $\left\{ t \in E \mid x(t) \geqslant a\right\}$ are closed, and the sets $\left\{ t \in E \mid x(t) < a\right\}$ and $\left\{ t \in E \mid x(t) > a\right\}$ are open in the set $E$ , regarded as a subspace of the metric space $X$ .

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Problem 3.18.65

Prove the stated assertion. If $x(t)$ is a function continuous on a subset $E$ of a metric space $X$ , then for any numbers $a \in R, b \in R, a < b$ , the set $\left\{ t \in E \mid a < x(t) < b\right\}$ is a set open in $E$ .

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Problem 3.18.66

Prove the stated assertion. The parallelepiped

\left\{ x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in l_{p}|| x_{n} \mid < 1, n = 1,2, \ldots \right\}

is open, while the parallelepiped

\left\{ x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in l_{p}|| x_{n} \mid \leqslant 1, n = 1,2, \ldots \right\}

is a set closed in the space $l_{p}$ , $1 \leqslant p < +\infty$ (see problem 5).

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Problem 3.18.67

Prove the stated assertion. If $x(t) \in C[a ; b]$ (see 31), then the set of functions

\left\{ y(t) \in G[a ; b] \mid \forall t \in [a ; b] y(t) \leqslant x(t)\right\}

is closed in the space $C[a ; b]$ .

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Problem 3.18.68

(1)(2)

Will the set of polynomials of the following degree be closed in the space $C[a ; b]$ (see 31):

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(1)

degree $\leqslant n$ ;

(2)

degree $= n$ ?

Problem 3.18.69

Prove the stated assertion. If $A$ and $B$ are closed (open) sets in the metric spaces $X$ and $Y$ respectively, then $A \times B$ is a closed (open) set in the product $X \times Y$ of the metric spaces $X$ and $Y$ (see problem 17).

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Problem 3.18.70

Prove the stated assertion. If a connected set contains more than one point, then it has no isolated points.

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Problem 3.18.71

Prove the stated assertion. The union of two intersecting connected sets is a connected set.

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Problem 3.18.72

Prove the stated assertion. For the union $A \cup B$ of two connected sets $A$ and $B$ of a metric space to be connected, it is necessary and sufficient that $(\bar{A} \cap B) \cup (A \cap \bar{B}) \neq \varnothing$ .

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Problem 3.18.73

Prove the stated assertion. If $A$ and $B$ are nonempty open (closed) sets of a metric space and $\bar{A} \cap \bar{B} = \varnothing$ , then the set $A \cup B$ is disconnected.

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Problem 3.18.74

Prove the stated assertion. If the union and the intersection of two closed sets of a metric space are connected, then both of these sets are connected.

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Problem 3.18.75

Prove the stated assertion. The union of an increasing sequence of connected sets is a connected set.

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Problem 3.18.76

Prove the stated assertion. The union of any collection of connected sets having a nonempty intersection is a connected set.

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Problem 3.18.77

Prove the stated assertion. If for any two points of a set $A$ there exists a connected set containing these points and contained in the set $A$ , then $A$ is a connected set.

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Problem 3.18.78

Prove the stated assertion. The closure of a connected set is connected.

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Problem 3.18.79

Prove the stated assertion. If $A$ is a connected set and $A \subset B \subset \bar{A}$ , then $B$ is also a connected set.

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Problem 3.18.80

Prove the stated assertion. A subset of the real line is connected if and only if it is a finite or infinite interval.

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Problem 3.18.81

Prove the stated assertion. The set of points of the plane, both of whose coordinates are rational, is disconnected.

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Problem 3.18.82

Prove the stated assertion. The set of points of the plane, at least one of whose coordinates is irrational, is a connected set.

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Problem 3.18.83

Prove the stated assertion. For the set $A \times B$ , lying in the product $X \times Y$ of the metric spaces $X$ and $Y$ (see problem 17 $), A \subset X, B \subset X$ , to be connected, it is necessary and sufficient that both sets $A$ and $B$ be connected.

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Problem 3.18.84

Prove the stated assertion. The space $R^{n}$ is a domain.

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Problem 3.18.85

Prove the stated assertion. An open ball in the space $R^{n}$ is a domain.

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Problem 3.18.86

Prove the stated assertion. An open ball in the space $l_{\infty }$ is a domain (see problem 4).

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Problem 3.18.87

Prove the stated assertion. If a sequence of points of a metric space converges, then it is fundamental.

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Problem 3.18.88

Prove the stated assertion. If some subsequence of a fundamental sequence converges, then the whole sequence also converges, and to the same limit as the given subsequence.

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Problem 3.18.89

Prove the stated assertion. The following spaces are complete:

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(1)

$R^{n}$ ;

(2)

$B(E)$ (see example 1);

(3)

$B_{C}(E)$ (see problem 3, part 4 $)$ );

(4)

$C B(E), E \subset R^{n}$ , consisting of all bounded functions continuous on $E$ , with the metric of the space $B(E)$ ;

(5)

$l_{\infty }$ (see problem 4);

(6)

$l_{p}, 1 \leqslant p < +\infty$ (see problem 5 $)$ .

Problem 3.18.90

Prove the stated assertion. The set $l_{\infty }^{(0)}$ of all sequences of real numbers $x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ tending to zero, with the metric

\rho (x ; y) = \max _{n}\left|x_{n}-y_{n}\right|, \quad x = \left(x_{1} ; \ldots ; x_{n} \ldots \right), \quad y = \left(y_{1} ; \ldots ; y_{n} ; \ldots \right) \in l_{\infty }^{(0)}

is a complete space.

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Problem 3.18.91

Prove the stated assertion. The metric space $S$ of all sequences of real numbers $x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ with the metric

$\rho (x ; y) = \sum_{n = 1}^{\infty } \frac{1}{2^{n}} \frac{\left|x_{n}-y_{n}\right|}{1+\left|x_{n}-y_{n}\right|}, \quad x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right), y = \left(y_{1} ; \ldots ; y_{n} ; \ldots \right) \in S$ , is complete.

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Problem 3.18.92

Prove the stated assertion. The space $C[a ; b]$ (see problem 31) is complete.

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Problem 3.18.93

Prove the stated assertion. The space $C^{n}[a ; b]$ of all functions having, on the segment $[a ; b]$ , a continuous derivative of order $n$ , with the metric

\rho (x ; y) = \sum _{k = 0}^{n} \max _{[a ; b]}\left|x^{(k)}(t)-y^{(k)}(t)\right|, \quad x(t), \quad y(t) \in C^{n}[a ; b]

is complete.

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Problem 3.18.94

Prove the stated assertion. The space $C^{\infty }[a ; b]$ of all functions infinitely differentiable on the segment $[a ; b]$ , with the metric

\rho (x ; y) = \sum _{n = 0}^{\infty } \frac{1}{2^{n}} \frac{\max _{[a ; b]}\left|x^{(n)}(t)-y^{(n)}(t)\right|}{1+\max _{[a ; b]}\left|x^{(n)}(t)-y^{(n)}(t)\right|}, \quad x(t), \quad y(t) \in C^{\infty }[a ; b]

is complete.

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Problem 3.18.95

Prove the stated assertion. A function $x(t)$ is said to satisfy the Hölder condition of degree $\alpha$ on the segment $[a ; b]$ if there exists a constant $c > 0$ such that for all $t_{1}, t_{2} \in [a ; b]$ the inequality $\left|x\left(t_{2}\right)-x\left(t_{1}\right)\right| \leqslant c\left|t_{2}-t_{1}\right|^{\alpha }$ holds. The space $H^{\alpha }[a ; b]$ of all functions satisfying the Hölder condition of degree $\alpha > 0$ on the segment $[a ; b]$ , with the metric

\rho (x ; y) = \max _{[a ; b]}\left|x(t)-y(t)\right|+\sup _{a \leqslant t_{1} < t_{2} \leqslant b} \frac{\left|\left[x\left(t_{2}\right)-y\left(t_{2}\right)\right]-\left[x\left(t_{1}\right)-y\left(t_{1}\right)\right]\right|}{\left|t_{2}-t_{1}\right|^{\alpha }}

is complete.

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Problem 3.18.96

Prove the stated assertion. The space $C(R)$ of all functions continuous on the real line $R$ , with the metric

\rho (x ; y) = \sum _{n = 1}^{\infty } \frac{1}{2^{n}} \frac{\max _{[-n ; n]}\left|x(t)-y(t)\right|}{1+\max _{[-n ; n]}\left|x(t)-y(t)\right|}, \quad x, y \in C(R)

is complete.

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Problem 3.18.97

Prove the stated assertion. The subspace of continuously differentiable functions of the space $C[a ; b]$ (see problem 31) is not complete.

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Problem 3.18.98

Prove the stated assertion. The space $C L_{1}[a ; b]$ (see problem 3.5) is not complete.

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Problem 3.18.99

Prove the stated assertion. The space $C L_{p}[a ; b], 1 \leqslant p < +\infty$ (see problem 7), is not complete.

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Problem 3.18.100

Denote by $C B(a ; b)$ the subspace of the space $B(a ; b)$ (see example 1) of bounded functions on the finite or infinite interval $(a ; b),-\infty \leqslant a < b \leqslant +\infty$ , consisting of all continuous functions. Will the space $C B(a ; b)$ be complete?

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Problem 3.18.101

If a metric space $X$ is complete with the metric $\rho (x ; y)$ , will it be complete with the metrics $\rho_{1}(x ; y), \rho_{2}(x ; y)$ , and $\rho_{3}(x ; y)$ of problem 18?

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Problem 3.18.102

Prove the stated assertion. In a complete metric space, a closed subset is complete.

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Problem 3.18.103

Prove the stated assertion. A complete subspace of a metric space is closed.

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Problem 3.18.104

Prove the stated assertion. If a set $A$ is dense in a metric space $X$ and $A \subset B \subset X$ , then the set $B$ is also dense in $X$ .

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Problem 3.18.105

Prove the stated assertion. If sets are dense in a metric space, then their union is also dense in this space.

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Problem 3.18.106

Give an example of two sets dense in a metric space whose intersection is not dense in it.

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Problem 3.18.107

(1)(2)(3)

Will the set of all polynomials in the space $C[a ; b]$ (see problem 31):

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(1)

open;

(2)

closed;

(3)

dense?

Problem 3.18.108

Prove the stated assertion. The set of continuous piecewise linear functions is dense in the space $C[a ; b]$ (see problem 31).

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Problem 3.18.109

(1)(2)(3)

Prove the stated assertion.

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(1)

The set of polynomials is dense in the space $C^{n}[a ; b]$ (see problem 93);

(2)

the set of polynomials is dense in the spaces $C L_{p}[a ; b], 1 \leqslant p < +\infty$ (see problem 7 $)$

(3)

the set of trigonometric polynomials is dense in the spaces of continuous periodic functions of period $2 \pi$ with the metrics

\begin{gathered} \rho (x ; y) = \max _{[0 ; 2 \pi ]}\left|x(t)-y(t)\right|, \\ \rho (x ; y) = \left(\int _{0}^{2 \pi }|x(t)-y(t)|^{p} d t\right)^{1 / p}, 1 \leqslant p < +\infty .\end{gathered}

Problem 3.18.110

Prove the stated assertion. In a complete metric space, the closure of any of its subspaces is a closed set.

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Problem 3.18.111

A numerical sequence is called finitely supported if only finitely many of its terms are nonzero.

Prove that the subspace of finitely supported sequences of the space $l_{\infty }^{(0)}$ (see problem 90) is not complete. Is either of the spaces $l_{\infty }^{(0)}$ and $l_{\infty }$ its completion?

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Problem 3.18.112

Will the subspace of finitely supported sequences (see problem 111) of the space $l_{p}, 1 \leqslant p < +\infty$ (see problem 5), be dense in this space?

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Problem 3.18.113

Let $(a ; b)$ be a finite or infinite interval: $-\infty \leqslant a < b \leqslant +\infty$ . A function $x(t)$ is called finitely supported (compactly supported) on the interval $(a ; b)$ if its support¹ is a compact set contained in the interval $(a ; b): \operatorname {supp} x \subset (a ; b)$ .

Will the set $C_{0}^{\infty }(a ; b)$ of infinitely differentiable functions with compact support on the interval $(a ; b)$ be dense in the space:

$C L_{p}(a ; b), \quad 1 \leqslant p < +\infty$ (see problem 7 $) ; C B(a ; b)$ (see problem 89 $)$ ?

The support of a function is the closure of the set of points at which the function is not equal to zero. The support of a function $f(x)$ is denoted $\operatorname {supp} f(x)$ . ↩

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Problem 3.18.114

Will the set of continuous functions with compact support (see problem 113) on the interval $(a ; b),-\infty \leqslant a < b \leqslant +\infty$ , be dense in the space $C B(a, b)$ of bounded functions continuous on the interval $(a, b)$ (see problem 100)?

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Problem 3.18.115

(1)(2)(3)

Prove the stated assertion. Two fundamental sequences $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ and $\left(y_{1} ; \ldots ; y_{n} ; \ldots \right)$ of a metric space $X$ are called equivalent if

\lim _{n \rightarrow \infty } \rho \left(x_{n} ; y_{n}\right) = 0

Let $X^{*}$ be the set of all classes of mutually equivalent fundamental sequences of the space $X$ . Then, if $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in x^{*}, \left(y_{1} ; \ldots ; y_{n} ; \ldots \right) \in y^{*}$ , then:

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(1)

the numerical sequence $\rho \left(x_{n} ; y_{n}\right)$ is fundamental and, consequently, $\lim_{n \rightarrow \infty } \rho \left(x_{n} ; y_{n}\right)$ exists;

(2)

this limit does not depend on the choice of the sequences $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in x^{*},\left(y_{1} ; \ldots ; y_{n} ; \ldots \right) \in y^{*}$ ;

(3)

the function

\rho ^{*}\left(x^{*} ; y^{*}\right) = \lim _{n \rightarrow \infty } \rho \left(x_{n} ; y_{n}\right),\left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in x^{*}, \quad \left(y_{1} ; \ldots ; y_{n} ; \ldots \right) \in y^{*}

is a metric on the set $X^{*}$ .

Problem 3.18.116

Prove the stated assertion. The mapping $f$ , assigning to each point $x \in X$ the class of equivalent fundamental sequences $x^{*}$ (see problem 115) containing the stationary sequence $\left\{ x ; x ; \ldots ; x ; \ldots \right\}$ , is an isometric mapping of $X$ into $X^{*}$ .

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Problem 3.18.117

Prove the stated assertion. If $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in x^{*}$ (see problem 115) and $f\left(x_{n}\right) = x_{n}^{*}$ (see problem 116), then $\lim_{n \rightarrow \infty } \rho^{*}\left(x_{n}^{*} ; x^{*}\right) = 0$ and the set $f(X)$ is dense in the space $X^{*}$ .

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Problem 3.18.118

Prove the stated assertion. The space $X^{*}$ (see problem 115) is complete.

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Problem 3.18.119

Prove the stated assertion. Every metric space has a completion.

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Problem 3.18.120

Prove the stated assertion. The completion of a metric space is unique up to isometric mappings.

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Problem 3.18.121

Prove the stated assertion. A sequence of nonempty sets of a metric space is called a Cauchy sequence if these sets are successively nested in one another and their diameters $\rightarrow 0$ .

If a sequence $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ of points of a metric space is fundamental, then the sequence of sets $E_{n} = \left(x_{n} ; x_{n+1} ; \ldots \right), n = 1,2, \ldots$ , is a Cauchy sequence.

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Problem 3.18.122

Prove the stated assertion. In a complete metric space, every Cauchy sequence (see problem 121) of closed sets has a nonempty intersection consisting of a single point.

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Problem 3.18.123

Prove the stated assertion. A metric space is complete if and only if every Cauchy sequence (see problem 121) of its subsets has a unique point that is a point of closure for all sets of the sequence.

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Problem 3.18.124

Construct an example of a sequence of closed balls of some metric space, successively nested in one another, whose diameters tend to zero but whose intersection is empty.

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Problem 3.18.125

Construct an example of a sequence of closed sets, successively nested in one another, in a complete metric space, whose intersection is empty.

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Problem 3.18.126

Prove the stated assertion. If $X$ is a complete metric space, $X = \bigcup_{n = 1}^{\infty } F_{n}$ , and all the sets $F_{n}$ are closed, then at least one of them contains an open ball.

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Problem 3.18.127

Prove the stated assertion. The spaces $l_{\infty }^{(0)}$ (see problem 90 $), l_{p}, 1 \leqslant p < +\infty$ (see problem 5), $C L_{p}[a ; b], 1 \leqslant p < +\infty$ (see problem 7) are separable.

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Problem 3.18.128

Prove the stated assertion. The spaces $l_{\infty }$ (see problem 4) and $B[a ; b]$ (see example 1) are not separable.

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Problem 3.18.129

Prove the stated assertion. A system $\Omega = \left\{ G_{\alpha }\right\} , \alpha \in \mathbf{U}$ ( $\mathbf{U}$ some index set), of open sets of a metric space $X$ is called a base for $X$ if every open set of the space $X$ is a union of sets of the system $\Omega$ contained in it.

A metric space is separable if and only if it has a countable base.

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Problem 3.18.130

Prove the stated assertion. If a metric space does not have a countable base, then for some $\varepsilon > 0$ there is an uncountable set of elements, the mutual distance between which is not less than $\varepsilon$ . Is the converse true?

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Problem 3.18.131

Prove the stated assertion. A compact set is a closed set in any metric space containing it.

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Problem 3.18.132

Prove the stated assertion. A closed subset of a compact set is a compact set.

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Problem 3.18.133

Prove the stated assertion. For two sets $A$ and $B$ of a metric space $X$ , the quantity $\rho (A ; B) = \inf_{x \in A, y \in B} \rho (x ; y)$ is called the distance between the sets $A$ and $B$ (see problem 15).

If $A$ and $B$ are closed disjoint sets, at least one of which is a compact set, then they are at a positive distance from each other.

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Problem 3.18.134

Give an example of two closed disjoint sets whose distance from each other equals zero.

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Problem 3.18.135

Prove that if $F_{1}$ and $F_{2}$ are closed disjoint sets in a metric space, then there exist disjoint open sets $G_{1} \supset F_{1}$ and $G_{2} \supset F_{2}$ .

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Problem 3.18.136

Construct an example of a closed set $F$ in some metric space $X$ and a point $x \in X$ such that for every point $y \in F$ the inequality holds

\rho (x ; y) > \rho (x ; F) = \inf _{z \in F} \rho (x ; z)

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Problem 3.18.137

Prove that if $F$ is a compact set in a metric space $X$ and $x \in X$ , then there exists a point $y \in F$ nearest to the point $x$ , i.e. such that $\rho (x ; F) = \rho (x ; y)$ .

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Problem 3.18.138

Give an example of a complete metric space $X$ , a set $A$ closed in it, and a point $x \in X$ such that in $A$ there is no point nearest to $x$ (see problem 137).

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Problem 3.18.139

Prove the stated assertion. For a function $x(t):[a ; b] \rightarrow R$ continuous on the segment $[a ; b]$ and any natural number $n$ , there exists a polynomial $P$ of best approximation of degree $n$ , i.e. a polynomial $P$ of degree at most $n$ such that, whatever polynomial $Q$ also of degree at most $n$ is taken, the inequality holds

\max _{[a ; b]}\left|x(t)-P(t)\right| \leqslant \max _{[a ; b]}\left|x(t)-Q(t)\right|.

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Problem 3.18.140

Prove the stated assertion. If $F_{1}$ and $F_{2}$ are compact sets in a metric space $X$ , then there exist points $x_{1} \in F_{1}$ and $x_{2} \in F_{2}$ such that $\rho \left(x_{1} ; x_{2}\right) = \rho \left(F_{1} ; F_{2}\right)$ (see problem 133). Is the compactness condition on both sets essential?

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Problem 3.18.141

Prove the stated assertion. Every totally bounded set is bounded.

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Problem 3.18.142

Prove the stated assertion. In the space $l_{2}$ (see example 2), the closed ball of radius 1 centered at the point $(0 ; 0 ; \ldots ; 0 ; \ldots )$ is bounded, but not totally bounded and not a compact set.

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Problem 3.18.143

Prove the stated assertion. If a set is totally bounded in some metric space, then for this set, for every $\varepsilon > 0$ , there exists a finite $\varepsilon$ -net consisting only of its points.

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Problem 3.18.144

Prove the stated assertion. The space $l_{2}$ (see example 2) is not isometric to any finite-dimensional space $R^{n}$ .

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Problem 3.18.145

Prove the stated assertion. Every set bounded in $R^{n}$ is also totally bounded.

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Problem 3.18.146

Prove the stated assertion. The set $Q^{\infty }$ of points $x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ of the space $l_{2}$ (see example 2), whose coordinates satisfy the inequalities $0 \leqslant x_{n} \leqslant 1 / 2^{n}$ , is a totally bounded set. (The set $Q^{\infty }$ is called the Hilbert brick.)

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Problem 3.18.147

Prove the stated assertion. A set in a metric space is totally bounded if and only if every sequence of this set contains a fundamental (Cauchy) subsequence.

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Problem 3.18.148

Prove the stated assertion. A complete totally bounded set in a metric space is a compact set.

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Problem 3.18.149

Prove the stated assertion. A metric space is a compact set if and only if it is totally bounded and complete.

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Problem 3.18.150

Prove the stated assertion. A compact set is a bounded set in any metric space containing it.

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Problem 3.18.151

Prove the stated assertion. For a subset of a complete metric space to be a compact set, it is necessary and sufficient that it be closed and totally bounded.

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Problem 3.18.152

Prove the stated assertion. The Hilbert brick (see problem 146) is a compact set.

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Problem 3.18.153

Prove the stated assertion. For a set lying in a complete metric space to be precompact, it is necessary and sufficient that it be totally bounded.

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Problem 3.18.154

Prove the stated assertion. A compact set is a separable metric space.

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Problem 3.18.155

Prove the stated assertion. From every cover of a separable metric space by open sets, a countable cover can be extracted.

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Problem 3.18.156

Prove the stated assertion. For a separable metric space to be a compact set, it is necessary and sufficient that every sequence of nonempty closed sets, successively nested in one another, have a nonempty intersection.

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Problem 3.18.157

Prove the stated assertion. For a separable metric space to be a compact set, it is necessary and sufficient that from any of its covers by open sets a finite cover can be extracted.

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Problem 3.18.158

Prove the stated assertion. For the product $X \times Y$ of metric spaces $X$ and $Y$ (see problem 17) to be a compact set, it is necessary and sufficient that the spaces $X$ and $Y$ be compact sets.

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Problem 3.18.159

Prove the stated assertion. If $A$ and $B$ are precompact sets of a metric space $X$ , then the set of numbers $\rho (x ; y)$ , where $x \in A, y \in B$ , is bounded.

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Problem 3.18.160

Prove the stated assertion. The intersection of any collection of compact sets is a compact set.

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Problem 3.18.161

Prove the stated assertion. The union of a finite collection of compact sets is a compact set.

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Problem 3.18.162

Prove the stated assertion. The intersection of a sequence of continua, successively nested in one another, is a continuum.

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Problem 3.18.163

Prove the stated assertion. If $F_{n}, n = 1,2, \ldots$ , is a sequence of compact sets in a metric space $X, F_{1} \supset F_{2} \supset \ldots \supset F_{n} \supset \ldots$ and $F = \bigcap_{n = 1}^{\infty } F_{n}$ , then for every $\varepsilon > 0$ there exists an index $n_{\varepsilon }$ such that for all $n > n_{\varepsilon }$ the inclusion holds

F_{n} \subset U(F ; \varepsilon ) \stackrel{\text{ def }}{ = } \bigcup _{x \in F} U(x ; \varepsilon )

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Problem 3.18.164

Prove the stated assertion. If $F_{n}, n = 1,2, \ldots$ , is a sequence of compact sets in a metric space $X, F_{1} \supset F_{2} \supset \ldots \supset F_{n} \supset \ldots$ , then for the intersection $F = \bigcap_{n = 1}^{\infty } F_{n}$ to consist of a single point, it is necessary and sufficient that $\lim_{n \rightarrow \infty } \operatorname {diam} F_{n} = 0$ .

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Problem 3.18.165

Prove the stated assertion. If $F_{n}, n = 1,2, \ldots$ , is a sequence of compact sets in a metric space such that the intersection of any finite collection of these compact sets is nonempty, then the intersection of all these compact sets is also nonempty.

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Problem 3.18.166

Is the following statement true: from any cover of a compact set by closures of open sets, a finite cover can be extracted?

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Problem 3.18.167

Prove that, whatever the cover $\left\{ G_{\alpha }\right\} , \alpha \in \mathbf{U}$ , of a compact set $X$ by open sets $G_{\alpha }$ , there exists a number $\delta > 0$ such that for every set $E \subset X$ whose diameter is less than $\delta$ , there is an element $G_{\alpha_{0}}$ of the given cover containing the set $E: G_{\alpha_{0}} \supset E$ .

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Problem 3.18.168

Prove the stated assertion ( $X, Y, Z$ are metric spaces). The definitions of continuity of a mapping $f: X \rightarrow Y$ given above in terms of sequences and in terms of neighborhoods are equivalent.

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Problem 3.18.169

Prove the stated assertion ( $X, Y, Z$ are metric spaces). A mapping $f: X \rightarrow Y$ is continuous at the point $x_{0} \in X$ if and only if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for every point $x \in X$ for which $\rho \left(x ; x_{0}\right) < \delta$ , the inequality $\rho \left(f(x) ; f\left(x_{0}\right)\right) < \varepsilon$ holds.

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Problem 3.18.170

Prove the stated assertion ( $X, Y, Z$ are metric spaces). A mapping $f: X \rightarrow Y$ is continuous on $X$ if and only if the preimage of every set open in $Y$ is a set open in $X$ .

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Problem 3.18.171

Prove the stated assertion ( $X, Y, Z$ are metric spaces). A mapping $f: X \rightarrow Y$ is continuous on $X$ if and only if the preimage of every set closed in $Y$ is a set closed in $X$ .

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Problem 3.18.172

(1)(2)

Is it necessarily true that, under a continuous mapping of one metric space into another:

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(1)

the image of every open set is open;

(2)

the image of a closed set is closed?

Problem 3.18.173

$X, Y, Z$ are metric spaces. Prove that the composition $g \circ f$ of continuous mappings $f: X \rightarrow Y$ and $g: \quad Y \rightarrow Z$ is a continuous mapping.

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Problem 3.18.174

(1)(2)

Will the mapping $f$ of the space $C[0 ; 1]$ (see problem 31) into itself be continuous, if it is given by the formula:

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(1)

$f(x) = \varphi x$ , where $\varphi = \varphi (t)$ is a fixed function from $C[0 ; 1]$ , and $x = x(t) \in C[0,1]$

(2)

$f(x) = x^{2}, x(t) = C[0 ; 1]$ ?

Problem 3.18.175

Will the mapping $f(x) = x^{2}$ of the space $C L_{1}[0 ; 1]$ (see problem 3.5) into itself be continuous?

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Problem 3.18.176

Let $X = \left\{ x \in C^{1}[0 ; 1], x(0) = 0\right\}$ (see problem 93). Will the mapping $f: X \rightarrow C[0 ; 1]$ , given by the formulas $f(x)(t) = \frac{x(t)}{t}, t \in (0 ; 1], f(x)(0) = \lim_{t \rightarrow 0} \frac{x(t)}{t}$ , be continuous?

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Problem 3.18.177

(1)(2)(3)

Are the following mappings of the space $C[0 ; 1]$ (see problem 31) into itself continuous:

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(1)

$f(x)(t) = \int_{0}^{t} x(s) d s$ ;

(2)

$f(x)(t) = \int_{0}^{t} x^{2}(s) d s ;$

(3)

$f(x)(t) = \int_{0}^{t} \sin (t-s) x(s) d s, t \in [0 ; 1]$ ?

Problem 3.18.178

(1)(2)(3)

Are the following mappings $f$ of the space $C L_{2}[0 ; 1]$ (see problem 3, part 8) into itself continuous:

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(1)

$f(x)(t) = \varphi (t) x(t)$ , where $\varphi (t)$ is a fixed function continuous on the segment $[0 ; 1]$ ;

(2)

$f(x)(t) = x^{2}(t)$ ;

(3)

$f(x)(t) = \int_{0}^{t} x(s) d s, t \in [0 ; 1]?$

Problem 3.18.179

(1)(2)

Will the mapping $f(x) = x(1)$ be continuous, if it is considered as a mapping:

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(1)

of $C[0 ; 1]$ (see problem 31) into $R$ ;

(2)

of $C L_{2}[0 ; 1]$ (see problem 3, part 8) into $R$ ?

Problem 3.18.180

Prove that if $f$ is a continuous mapping of a space $X$ onto a space $Y$ and $E$ is a set dense in $X$ , then $f(E)$ is a set dense in $Y$ .

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Problem 3.18.181

Prove that if $f: X \rightarrow Y$ is a continuous mapping of $X$ into $Y$ and $X$ is a compact set, then $f$ is uniformly continuous on $X$ .

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Problem 3.18.182

Is differentiation, regarded as a mapping of the subspace consisting of continuously differentiable functions of the space $C[a ; b]$ (see problem 31) into the space $C[a ; b]$ itself, continuous?

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Problem 3.18.183

Is differentiation, regarded as a mapping of the space $C^{1}[a ; b]$ (see problem 93) onto the space $C[a ; b]$ (see problem 31), continuous?

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Problem 3.18.184

Let $E$ be a set dense in a metric space $X$ , and let $f: E \rightarrow Y$ be a uniformly continuous mapping of $E$ into a complete metric space $Y$ . Prove that there exists a unique mapping $f^{*}: X \rightarrow Y$ whose restriction to $E$ coincides with $f$ .

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Problem 3.18.185

Give an example of a continuous mapping defined on a dense subset of a metric space that has no continuous extension to the whole space.

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Problem 3.18.186

Prove the stated assertion. For a sequence of mappings $f_{n}: X \rightarrow Y$ , $n = 1,2, \ldots$ , to converge uniformly to some mapping of $X$ into $Y$ , where $Y$ is a complete space, it is necessary and sufficient that for every $\varepsilon > 0$ there exist a number $n_{\varepsilon }$ such that for all $n > n_{\varepsilon }, m > n_{\varepsilon }$ and all points $x \in X$ the inequality $\left|f_{n}(x)-f_{m}(x)\right| < \varepsilon$ holds.

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Problem 3.18.187

Prove the stated assertion. If a sequence of mappings $f_{n}: X \rightarrow Y$ converges uniformly to a mapping $f: X \rightarrow Y$ and all the mappings $f_{n}$ are continuous at a point $x_{0} \in X$ , then the mapping $f$ is also continuous at that point.

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Problem 3.18.188

Prove the stated assertion. A contraction mapping $f: X \rightarrow X$ is uniformly continuous on $X$ .

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Problem 3.18.189

Prove the stated assertion. A contraction mapping $f$ of a complete metric space into itself has a unique fixed point, i.e., a point such that $f(x) = x$ .

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Problem 3.18.190

Prove the stated assertion. If some power of a mapping of a complete metric space into itself is a contraction mapping, then the mapping itself has a unique fixed point.

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Problem 3.18.191

(1)(2)

Prove the stated assertion. If the function $f(t)$ is continuous on the segment $[a ; b]$ , the function $K(t ; s)$ is continuous on the square $Q = [a ; b] \times [a ; b], c = \max_{Q}\left|K(t ; s)\right|, \lambda$ is some number, and then:

A(x) = \lambda \int _{a}^{t} K(t ; s) x(s) d s+f(t)

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(1)

the mapping $A$ maps the space $C[a ; b]$ (see problem 31) into itself;

(2)

for any functions $x_{1} \in C[a ; b], x_{2} \in C[a ; b]$ and any $t \in [a ; b]$ the inequality

\left|A^{n}\left(x_{1}\right)(t)-A^{n}\left(x_{2}\right)(t)\right| \leqslant \frac{\lambda ^{2} c^{n}(b-a)^{n}}{n !} \max _{[a ; b]}\left|x_{1}(t)-x_{2}(t)\right|

holds.

Problem 3.18.192

Prove the stated assertion. The mapping $A(x)$ of the preceding problem, of the space $C[a ; b]$ into itself, has a unique fixed point, i.e., the equation

x(t) = \lambda \int _{a}^{t} K(t ; s) s(s) d s+f(t)

has, for any $\lambda$ , a unique continuous solution.

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Problem 3.18.193

Give an example of a mapping $f$ of a complete metric space $X$ into itself for which, for any two points $x \in X$ , $y \in X$ , the condition $\rho (f(x) ; f(y)) \leqslant \rho (x ; y)$ holds, but which has no fixed point.

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Problem 3.18.194

Prove the stated assertion. Every continuous mapping of a segment into itself has a fixed point.

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Problem 3.18.195

Prove the stated assertion. Every increasing mapping of a segment into itself has a fixed point.

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Problem 3.18.196

Prove the stated assertion. The continuous image of a compact set is compact.

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Problem 3.18.197

Prove the stated assertion. A one-to-one continuous mapping of a compact set is a homeomorphism.

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Problem 3.18.198

Prove the stated assertion. Every continuous function $f: X \rightarrow R$ defined on a compact set $X$ is bounded on it and attains its greatest and least values.

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Problem 3.18.199

Prove the stated assertion. The continuous image of a connected set is a connected set.

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Problem 3.18.200

Prove the stated assertion. The continuous image of a continuum is a continuum.

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Problem 3.18.201

Prove the stated assertion. If a metric space is the continuous image of a separable metric space, then it is also a separable space.

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Problem 3.18.202

Prove the stated assertion. The set of functions $\left\{ x: x(t) = k t^{2}, 0 \leqslant k \leqslant 1\right\}$ is compact in the space $C[0 ; 1]$ .

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Problem 3.18.203

Prove the stated assertion. A family $S = \left\{ x\right\}$ of functions $x = x(t)$ of the space $C[a ; b]$ (see problem 31) is called uniformly bounded if there exists a constant $c > 0$ such that for all $x \in S$ and all $t \in [a ; b]$ the inequality $\left|x(t)\right| \leqslant c$ holds. A family $S = \left\{ x\right\}$ of functions $x = x(t)$ of the space $C[a ; b]$ is called equicontinuous if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $x \in S$ and all $t_{1}, t_{2} \in [a ; b]$ for which $\left|t_{2}-t_{1}\right| \leqslant \delta$ , the inequality $\left|x\left(t_{2}\right)-x\left(t_{1}\right)\right| < \varepsilon$ holds.

For a family $S = \left\{ x\right\}$ of functions $x = x(t)$ continuous on the segment $[a ; b]$ to be precompact in the space $C[a ; b]$ , it is necessary and sufficient that this family be uniformly bounded and equicontinuous (Arzelà's theorem).

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Problem 3.18.204

Is the set of all polynomials $P(t)$ of degree not exceeding a given natural number $n: P(t) = \sum_{k = 0}^{n} a_{k} t^{k}$ , for which $\left|a_{k}\right| \leqslant 1, k = 0,1,2, \ldots , n$ , compact in the space $C[a ; b]$ ?

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Problem 3.18.205

Prove the stated assertion. Every set bounded in the space $C^{1}[a ; b]$ (see problem 93) is precompact in the space $C[a ; b]$ (see problem 31).

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Problem 3.18.206

Prove the stated assertion. Let $C(X)$ be the space of continuous functions $x: X \rightarrow R$ on a compact set $X$ , with the metric

\rho (x ; y) = \max _{X}\left|x(t)-y(t)\right|, \quad x, y \in C(X)

In this case a set $E \subset C(X)$ is precompact if and only if it is uniformly bounded (i.e., there exists a constant $c > 0$ such that for all $x \in E$ and all $t \in X$ the inequality $\left|x(t)\right| \leqslant c$ holds) and uniformly continuous (i.e., for every $\varepsilon > 0$ there exists $\delta > 0$ such that for every function $x \in E$ , the inequality $\rho \left(t_{2} ; t_{1}\right) < \delta$ implies that $\left|x\left(t_{2}\right)-x\left(t_{1}\right)\right| < \varepsilon$ ).

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Problem 3.18.207

Prove the stated assertion. In the subspace of uniformly continuous functions of the space $B(X)$ (see example 1), where $X$ is a metric space, a set is precompact if and only if it is uniformly bounded and equicontinuous (see problem 206).

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Problem 3.18.208

Determine which of the families of functions listed below are precompact in the space $C[a ; b]$ (see problem 31), and for which $a$ and $b$ :

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(1)

$t^{n}, n = 1,2, \ldots$

(2)

$(\alpha t)^{n}, \alpha \in R, n = 1,2, \ldots$

(3)

$\sin n t, n = 1,2, \ldots$

(4)

$\sin (t+n), n = 1,2, \ldots$

(5)

$e^{t+\alpha }, \alpha \in R$

(6)

$e^{t-\alpha }, \alpha \in R, \alpha > 0$ .

Problem 3.18.209

Which of the sets listed below are compact in the space $C[0 ; 1]$ (see problem 31), where $c$ is some constant:

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(1)

$\left\{ x \in C[0 ; 1]|| x(t) \mid \leqslant c\right\}$

(2)

$\left\{ x \in C^{1}[0 ; 1]|| x(t) \mid \leqslant c,| x^{\prime }(t) \mid \leqslant c\right\}$ ;

(3)

$\left\{ x \in C^{2}[0 ; 1]|| x(t) \mid \leqslant c,| x^{\prime }(t) \mid \leqslant c,| x^{\prime \prime }(t) \mid \leqslant c\right\}$ ;

(4)

$\left\{ x \in C^{2}[0 ; 1]|| x(t) \mid \leqslant c,| x^{\prime \prime }(t) \mid \leqslant c\right\}$ ;

(5)

$\left\{ x \in C^{2}[0 ; 1]|| x^{\prime }(t) \mid \leqslant c,| x^{\prime \prime }(t) \mid \leqslant c\right\}$ ?

Problem 3.18.210

Prove: for a set $E \subset l_{p}, 1 \leqslant p < +\infty$ (see problem 5), to be precompact, it is necessary and sufficient that it be bounded and that for every $\varepsilon > 0$ there exist $n_{\varepsilon }$ such that for all $x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in E$ the inequality

\sum _{k = n_{\varepsilon }}^{\infty }\left|x_{k}\right|^{p} < \varepsilon

holds.

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Footnotes