Compact sets

Prove Theorem 4.1.

Theorem 4.1: in any metric space $X$, sequential compactness of a set $M$ is equivalent to compactness of $M$.

Let $U$ be an open cover of a metric compact $(X, \rho )$. Prove that there exists $r > 0$ such that every ball $B(x, r)$ is contained in some element of the cover $U$.

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

Prove that any subset of a precompact set is precompact.

Prove that in any finite-dimensional normed space precompactness is equivalent to boundedness.

Prove that in a discrete metric space, in the space $\mathbb {N}$, and in the space $\mathbb {Z}$, a set is precompact if and only if it consists of a finite number of points.

Prove that if a mapping $f: X \rightarrow Y$ of metric spaces is continuous and $X$ is compact, then $f(X)$ is compact in $Y$.

Prove that if a set $M$ is precompact in an incomplete metric space $X$, then it is precompact also in the completion of this space¹. Give a counterexample to the converse statement.

Definition 1.16: A metric space $Y$ is called a completion of a metric space $X$ if: (1) $Y$ is complete; (2) there is an isometric embedding $f: X \rightarrow Y$ such that the set $f(X)$ is everywhere dense in $Y$.

Give an example of an incomplete metric space $X$ and a totally bounded, but not precompact, set $M$ in it (compare this problem with Theorem 4.2).

Theorem 4.2 (F. Hausdorff, 1914): In any metric space, precompactness of a set $M$ implies total boundedness of $M$. In a complete metric space the converse also holds: every totally bounded set is precompact.

Prove that if a set $M$ in a metric space $X$ is totally bounded, then any sequence of points of the set $M$ contains a fundamental subsequence.

Prove that if for some $\varepsilon > 0$ a finite $\varepsilon$-net exists for a set $M$ of a metric space $X$, then there exists a finite $2 \varepsilon$-net consisting of points of the set $M$.

In the closed unit ball of the space $l_{p}, 1 \leqslant p \leqslant \infty$, exhibit a sequence containing no fundamental subsequence.

Prove that in any infinite-dimensional normed space $X$ any ball (closed or open) is not precompact.

Prove that any precompact set in an infinite-dimensional Banach space is nowhere dense.

Prove that if in a normed space $X$ a set $M$ is precompact, then it is bounded and can be approximated with any precision by a finite-dimensional subspace, i.e., for any $\varepsilon > 0$ there exists a finite-dimensional subspace $X_{0}$ such that $\operatorname {dist}_{0}\left(M, X_{0}\right) < \varepsilon$ (see Definition 1.17). Prove the converse statement for a Banach space $X$.

Definition 1.17: Let $X$ be a metric space, $A \subset X$ a nonempty set and $x \in X$. The distance from the point to the set is the number $\operatorname {dist}(x, A) := \inf \left\{ \rho (x, y): y \in A\right\}$. Here, as in the book's text before this problem, the distance between two sets is denoted $\operatorname {dist}_{0}(A, B) = \inf \left\{ \rho (x, y): x \in A, y \in B\right\}$.

Let $(X, \rho )$ be a metric space, and let $M$ be a precompact set in it. Prove that $(M, \rho )$ is a separable metric space.

Let $M$ be a compact set in a Banach space $X$. Prove that there exists a sequence $x_{n} \rightarrow 0$ such that $M \subset \overline{\operatorname {conv}\left\{ x_{n}\right\}_{1}^{\infty }}$.

Let $X$ be a metric space, and let $M$ be a boundedly compact set in it. Prove that for any point $x \in X$ the distance from $x$ to $M$ is attained, i.e., there exists $y \in M$ such that $\operatorname {dist}(x, M) = \rho (x, y)$.

Let $X$ be a metric space, and let $M_{1}$ and $M_{2}$ be compact sets in it. Prove that $\operatorname {dist}_{0}\left(M_{1}, M_{2}\right) = \rho \left(x_{0}, y_{0}\right)$ (see Definition 1.17) for some $x_{0} \in M_{1}$ and $y_{0} \in M_{2}$.

Definition 1.17: Let $X$ be a metric space, $A \subset X$ a nonempty set and $x \in X$. The distance from the point to the set is the number $\operatorname {dist}(x, A) := \inf \left\{ \rho (x, y): y \in A\right\}$. Here, as in the book's text before problem 4.15, the distance between two sets is denoted $\operatorname {dist}_{0}(A, B) = \inf \left\{ \rho (x, y): x \in A, y \in B\right\}$.

Let $X$ be a metric space, and let $M$ be a compact set in it. Prove that $\operatorname {diam}(M) = \rho (x, y)$ for some $x, y \in M$.

Let $X$ be a metric compact. Prove that every subset of $X$ has a Chebyshev center.

Prove that in a complete metric space any system of closed nonempty nested sets has nonempty intersection, if all these sets are contained in some compact set. ¹

Prove that the function $\operatorname {dist}_{\mathrm{H}}$ satisfies the axioms of a metric.

Denote by $\mathcal{N}(X)$ the family of all subsets of a metric space $X$ consisting of a finite number of points. Prove that if $X$ is complete, then the space $(\exp (X), \operatorname {dist}_{\mathrm{H}})$ is a completion of the space $(\mathcal{N}(X), \operatorname {dist}_{\mathrm{H}})$.

Let $X$ be a metric compact. Prove that the space $(\exp (X), \operatorname {dist}_{\mathrm{H}})$ is also a compact.

Let $X$ be a complete metric space, and let $\left\{ f_{j}\right\}_{1}^{n}$ be contraction mappings from $X$ to $X$. Define the mapping

Prove that $F$ is a contraction mapping in $\left(\exp (X), \operatorname {dist}_{\mathrm{H}}\right)$.

Prove that the following sets are fixed points for certain contraction mappings. Find these mappings.

Let $M$ be a set in a complete metric space $X$ such that every real-valued function continuous on $M$ is bounded. Prove that $M$ is a compact.

Let $M$ be a set in a complete metric space $X$ such that every real-valued function continuous and bounded on $M$ attains its supremum and infimum. Prove that $M$ is a compact.

Let $M$ be a compact set in a complete metric space $X_{1}, f: M \rightarrow X_{2}$, where $X_{2}$ is also a complete metric space. Prove that the continuity of the mapping $f$ implies its uniform continuity.

Let $M$ be a set in a complete metric space $X$ such that every real-valued function continuous on $M$ is uniformly continuous. Prove that $M$ is a compact.

Let $M$ be a compact in a metric space $X$, and let $Y$ be a normed space. Denote by $C(M, Y)$ the set of continuous functions from $M$ to $Y$. Prove that the norm $\left\| x\right\|_{C(M, Y)} = \max_{t \in M}\left\| x(t)\right\|_{Y}$ turns $C(M, Y)$ into a normed space (the linear operations are introduced pointwise), and if $Y$ is a Banach space, then the space $C(M, Y)$ is also a Banach space. ¹

(Dini, 1872). Let $M$ be a metric compact, and let the sequence of functions $\left\{ f_{n}: M \rightarrow Y\right\}_{1}^{\infty }$, continuously mapping $M$ into a normed space $Y$, converge to a continuous function $f$ pointwise, and for every $x \in M$

Prove that $f_{n} \rightarrow f$ in the space $C(M, Y)$. ¹

Prove that if a set $M$ is compact in a metric space $X$, and a mapping $f: M \rightarrow M$ satisfies the inequality $\rho (f(x), f(y)) < \rho (x, y)$ for any $x \neq y$, then there exists a unique fixed point $x \in M$ (compare with problem 1.74 and Theorem 1.4).

Theorem 1.4 (contraction mapping principle): let $f: X \rightarrow X$ be a contraction mapping (i.e., there exists $q \in [0, 1)$ such that $\rho (f(x_{1}), f(x_{2})) \leqslant q \cdot \rho (x_{1}, x_{2})$ for all $x_{1}, x_{2} \in X$) on a complete metric space $X$. Then there exists a unique point $x_{0} \in X$ such that $f(x_{0}) = x_{0}$; this point is the limit of the recursively defined sequence $x_{n} = f(x_{n-1})$ with an arbitrary starting point $x_{1} \in X$.

Prove that in a metric space any isometric embedding of a compact into itself is an isometry of this compact (and show that this is false for an arbitrary closed bounded set).

Let $X$ be a complete metric space, $M$ a compact set in it, $\varphi : M \rightarrow M$ a mapping satisfying the inequality $\rho (\varphi (x), \varphi (y)) \geqslant \rho (x, y)$. Prove that $\varphi$ is an isometry of the set $M$.

Find the approximative dimension:

Find the approximative dimension:

Let $M$ be an equicontinuous set in the space $C[0,1]$. Let a sequence of functions from $M$ converge pointwise to a continuous function $x$. Prove that this sequence also converges in norm to the same limit.

Which of the sets listed below are precompact in the space $C[0,1]$:

Is the set precompact in the space $C[0,1]$

Is the set precompact in the space $C[a, b]$

(V. D. Erokhin). Prove that every compact of diameter $d$ in the space $C[0,1]$ lies in some closed ball of radius $d / 2$ (i.e., its Chebyshev radius equals $d / 2$). Prove that for Hilbert spaces (of dimension greater than 1) this is false.

Prove that a set $M$ in the space $C^{n}[0,1], n = 1,2, \ldots$, is precompact if and only if it is bounded and the set $M_{n}$ of functions $\left\{ x^{(n)}(t): x \in M\right\}$ is equicontinuous.

Is it true that a set $M$ in the space $C^{n}[0,1], n = 1,2, \ldots$, is precompact if and only if the set $M_{n}$ from the previous problem is precompact in $C[0,1]$?

Give an example of a set $M$ of continuously differentiable functions, precompact in the space $C[0,1]$, but not precompact in the space $C^{1}[0,1]$.

Is the set of analytic functions precompact in the space $C^{1}[0,1]$

where the series converges in a disk $D \subset \mathbb {C}$ containing $[0,1]$?

Prove the statement:

Under what condition on the sequence $\left\{ a_{k}\right\}_{k = 1}^{\infty }$ is the parallelepiped $\Pi_{\left\{ a_{k}\right\} } = \left\{ x \in l_{p}:\left|x_{n}\right| \leqslant a_{n}\right\}$ precompact in the space $l_{p}(1 \leqslant p < \infty )$?

Under what condition on the sequence $\left\{ a_{k}\right\}_{k = 1}^{\infty }$ is the ellipsoid $E_{\left\{ a_{k}\right\} } = \left\{ x \in l_{2}: \sum_{n = 1}^{\infty } \frac{\left|x_{n}\right|^{2}}{a_{n}^{2}} \leqslant 1\right\}$ precompact in the space $l_{2}$?

Prove that any compact set $M \subset l_{2}$ is contained in some ellipsoid of the form $\left\{ x \in l_{2}: \sum_{n = 1}^{\infty } \frac{\left|x_{n}\right|^{2}}{a_{n}^{2}} \leqslant 1\right\}$, where $a_{n} \rightarrow 0$.

Prove the precompactness in the space $l_{1}$ of the set $M = \left\{ x: \sum_{n = 1}^{\infty } n\left|x_{n}\right| \leqslant 1\right\}$.

Is the set $M = \left\{ x: x_{n} = \int_{0}^{1} \frac{y(s)}{n^{2}+s^{2}} d s\right\}$ precompact in the space $l_{1}$, where the functions $y$ are chosen from the closed unit ball of the space $C[0,1]$?

For which values of the parameters $\alpha \in \mathbb {R}$ and $\beta > 0$ is the set $M_{\alpha , \beta } = \left\{ x \in l_{p}: \sum_{n = 1}^{\infty } n^{\alpha }\left|x_{n}\right|^{\beta } \leqslant 1\right\}$ precompact in the space $l_{p} (1 \leqslant p < \infty )$?

Is the set

precompact in the space $l_{1}$?

Let $H$ be a Hilbert space, $\left\{ e_{n}\right\}_{1}^{\infty }$ an orthonormal basis in it. Prove that a set $M$ is precompact in $H$ if and only if it is bounded and for every $\varepsilon > 0$ there exists $N \in \mathbb {N}$ such that $\sum_{k = N}^{\infty }\left|\left(x, e_{k}\right)\right|^{2} < \varepsilon^{2}$ for all $x \in M$.

Is the set $\left\{ \sin (\pi n t)\right\}_{n = 1}^{\infty }$ precompact in the space $L_{p}[0,1], p \in [1, \infty ]$?

Prove that if $\left\{ \varphi_{\alpha }\right\}_{\alpha \in \Lambda }$ is a precompact set of functions in $C[a, b]$, then the set $\left\{ x_{\alpha } = \varphi_{\alpha } \theta \right\}_{\alpha \in \Lambda }$, where $\theta$ is a fixed function from $L_{2}[a, b]$, is precompact in $L_{2}[a, b]$.

Prove that a set $M$ in the space $W_{p}^{n}[0,1], 1 \leqslant p < \infty , n = 1,2, \ldots$, is precompact if and only if it is bounded and for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $x \in M$ and $h \in [0, \delta ]$ the inequality holds

Determine whether the sets $M_{a} = \left\{ n^{-a} \sin (n t)\right\}_{n = 1}^{\infty }$ are precompact for $a \in [0,1]$ in the spaces:

Is the unit ball of the space $C^{1}[a, b]$ precompact in the space $C[a, b]$?

Prove the compactness of the embeddings:

Prove that the following embeddings of spaces are not compact: