Math Problems Vault

Problem 4.39

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.

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Problem 4.40

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

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(a)

$\left\{ t^{n}\right\}_{n \in \mathbb {N}}$ ;

(b)

$\left\{ \sin (t+n)\right\}_{n \in \mathbb {N}}$ ;

(c)

$\left\{ \sin \alpha t\right\}_{\alpha \in \mathbb {R}}$ ;

(d)

$\left\{ \sin \alpha t\right\}_{\alpha \in [0,1]}$ ;

(e)

$\left\{ \operatorname {arctg} \alpha t\right\}_{\alpha \in \mathbb {R}}$ ;

(f)

$\left\{ e^{t-\alpha }\right\}_{\alpha \geqslant 0}$ ;

(g)

$\left\{ t^{2 n}-t^{n}\right\}_{n \in \mathbb {N}}$ ;

(h)

$\left\{ t^{n+1}-t^{n}\right\}_{n \in \mathbb {N}}$ ;

(i)

$\left\{ \sqrt{1-t^{n}}\right\}_{n \in \mathbb {N}}$ ?

Problem 4.41

(a)(b)(c)

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

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(a)

$M_{1} = \left\{ x(t) = \int_{0}^{t} y(s) d s: y \in C[0,1],\left\| y\right\|_{C[0,1]} \leqslant 1\right\}$ ;

(b)

$M_{2} = \left\{ x(t) = \sum_{n = 1}^{\infty } \frac{b_{n}}{t^{2}+n}: \sum_{n = 1}^{\infty } \frac{\left|b_{n}\right|}{n} \leqslant 1\right\}$ ;

(c)

$M_{3} = \left\{ x(t) = \int_{0}^{1} \frac{y(s)}{t^{2}+s^{2}+1} d s: y \in C[0,1],\left\| y\right\|_{C[0,1]} \leqslant 1\right\}$ ?

Problem 4.42

(a)(b)(c)

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

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(a)

$M_{1} = \left\{ x(t) \in C^{1}[a, b]:\left|x(a)\right| \leqslant C_{1}, \int_{a}^{b}\left|x^{\prime }(t)\right| d t \leqslant C_{2}\right\}$ ;

(b)

$M_{2} = \left\{ x(t) \in C^{1}[a, b]:\left|x(a)\right| \leqslant C_{1}, \int_{a}^{b}\left|x^{\prime }(t)\right|^{2} d t \leqslant C_{2}\right\}$ ;

(c)

$M_{3} = \left\{ x(t) \in C^{1}[a, b]: \int_{a}^{b}\left(|x(t)|^{2}+\left|x^{\prime }(t)\right|^{2}\right) d t \leqslant C\right\}$ ?

Problem 4.43

(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.

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Problem 4.44

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.

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Problem 4.45

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 $C[0,1]$ ?

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Problem 4.46

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

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Problem 4.47

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

M = \left\{ x(t) \in C^{1}[0,1]:\left\| x\right\| _{C^{1}} < 1, x(t) = \sum _{k = 0}^{\infty } a_{k} t^{k}\right\}

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

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Problem 4.48

(a)(b)(c)(d)

Prove the statement:

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(a)

item 2;

(b)

item 3;

(c)

item 4;

(d)

item 5 from the list of precompactness criteria at the beginning of §4.2.

Problem 4.49

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 $l_{p}(1 \leqslant p < \infty )$ ?

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Problem 4.50

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 $l_{2}$ ?

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Problem 4.51

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 $a_{n} \rightarrow 0$ .

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Problem 4.52

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

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Problem 4.53

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 $C[0,1]$ ?

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Problem 4.54

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 $l_{p} (1 \leqslant p < \infty )$ ?

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Problem 4.55

Is the set

M = \left\{ x \in l_{1}: x_{n} = \int _{-\pi }^{\pi } y(t) \cos n t d t, y \in C^{1}[-\pi , \pi ],\left\| y\right\| _{C^{1}} \leqslant 1\right\}

precompact in the space $l_{1}$ ?

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Problem 4.56

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 $x \in M$ .

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Problem 4.57

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

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Problem 4.58

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 $L_{2}[a, b]$ .

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Problem 4.59

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

\left(\int _{0}^{1-h}\left|x^{(n)}(t+h)-x^{(n)}(t)\right|^{p} d t\right)^{1 / p} < \varepsilon

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Problem 4.60

(a)(b)(c)

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:

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(a)

$C[0,1]$ ,

(b)

$C^{1}[0,1] ;$

(c)

$W_{p}^{1}[0,1], p \in [1, \infty )$ .

Problem 4.61

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

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Problem 4.62

(a)(b)(c)(d)

Prove the compactness of the embeddings:

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(a)

$C^{n+1}[0,1] \Subset C^{n}[0,1]$ ;

(b)

$W_{2}^{1}[0,1] \Subset L_{2}[0,1]$ ;

(c)

$W_{2}^{1}[0,1] \Subset C[0,1]$ ;

(d)

$l_{2, n} \Subset l_{2}$ (see problem 3.13).

Problem 4.63

Prove that the following embeddings of spaces are not compact:

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(a)

$c_{0} \subset c \subset l_{\infty }$ ;

(b)

$l_{p_{1}} \subset l_{p_{2}} \subset c_{0}$ , where $1 \leqslant p_{1} < p_{2} \leqslant \infty$ ;

(c)

$C_{0}(\mathbb {R}) \subset B C(\mathbb {R})$ ;

(d)

$C[0,1] \subset L_{p}[0,1]$ , where $p \in [1, \infty ]$ ;

(e)

$L_{p_{1}}[0,1] \subset L_{p_{2}}[0,1]$ , where $1 \leqslant p_{2} < p_{1} < \infty$ .