Math Problems Vault

Problem 1.5

Find the norm of an element in a given space:

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(1)

$x(t) = t^{n}$ in $C(\left[0,1\right]), n \geq 1$ ;

(2)

$x(t) = e^{-t}$ in $C(\left[0,1\right])$ ;

(3)

$x = \left(1, \frac{1}{2}, \frac{1}{3}, \ldots \right)$ in $l_{2}$ ; in $l_{\infty }$ ;

(4)

$x(t) = t$ in $L_{4}(\left[0,1\right])$ ;

(5)

$x(t) = t^{2}$ in $C^{1}(\left[0,1\right])$ ;

(6)

$x(t) = \chi_{\mathbb {Q}}(t)$ in $L_{1}(\left[0,1\right]) ;$

(7)

$x(t) = \chi_{\mathbb {R} \backslash \mathbb {Q}}(t)$ in $L_{2}(\left[0,1\right])$ ;

(8)

$x(t) = 1-e^{t}$ in $L_{3}(\left[0,1\right])$ ;

(9)

$x(t) = \sin t+\cos t$ in $L_{2}(\left[0,1\right])$ ;

(10)

$x(t) = t \ln t$ in $L_{1}(\left[1, e\right])$ .

Problem 1.6

(1)(2)(3)

Are the following functions norms on $C(\left[a, b\right])$ ?

?

(1)

$\left\| x\right\| = \max_{a \leq t \leq \frac{1}{2}(a+b)}\left|x(t)\right|+\left|x(b)\right|$ ;

(2)

$\left\| x\right\| = \left(\int_{a}^{b} \alpha (t)\left|x(t)\right|^{2} d t\right)^{\frac{1}{2}}$ , where $\alpha \in C(\left[a, b\right])$ is such that $\alpha (t) > 0, t \in \left[a, b\right] ;$

(3)

$\left\| x\right\| = \max_{a \leq t \leq \frac{1}{2}(a+b)}\left|x(t)\right|+\int_{\frac{1}{2}(a+b)}^{b}\left|x(t)\right| d t$ .

Problem 1.7

Are the following functions norms on $C^{1}(\left[a, b\right])$ ?

?

(1)

$\left\| x\right\| = \max_{a \leq t \leq b}\left|x(t)\right|$ ;

(2)

$\left\| x\right\| = \max_{a \leq t \leq b}\left|x^{\prime }(t)\right|$ ;

(3)

$\left\| x\right\| = \max_{a \leq t \leq \frac{1}{2}(a+b)}\left|x(t)\right|+\max_{\frac{1}{2}(a+b) \leq t \leq b}\left|x^{\prime }(t)\right|$ ;

(4)

$\left\| x\right\| = \left|x(a)\right|+\max_{a \leq t \leq b}\left|x^{\prime }(t)\right|$ ;

(5)

$\left\| x\right\| = \left|x(b)-x(a)\right|+\max_{a \leq t \leq b}\left|x^{\prime }(t)\right|$ ;

(6)

$\left\| x\right\| = \int_{a}^{b}\left|x(t)\right| d t+\max_{a \leq t \leq b}\left|x^{\prime }(t)\right|$ .

Problem 1.8

(1)(2)(3)

Are the following functions norms on $C^{2}(\left[a, b\right])$ ?

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(1)

$\left\| x\right\| = \left|x(a)\right|+\left|x^{\prime }(a)\right|+\max_{a \leq t \leq b}\left|x^{\prime \prime }(t)\right|$ ;

(2)

$\left\| x\right\| = \left|x(a)\right|+\left|x(b)\right|+\max_{a \leq t \leq b}\left|x^{\prime \prime }(t)\right|$ ;

(3)

$\left\| x\right\| = \int_{a}^{b}\left|x(t)\right| d t+\left|x^{\prime }(a)\right|+\max_{a \leq t \leq b}\left|x^{\prime \prime }(t)\right|$ .

Problem 1.9

(1)(2)(3)(4)

Is the sequence of elements convergent in the given space? If so, find its limit.

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(1)

In the space $C(\left[0,1\right])$ :

(1) $x_{n}(t) = t^{n}$ ;

(2) $x_{n}(t) = t^{n}-t^{n+1}$ ;

(3) $x_{n}(t) = t^{n}-t^{2 n}$ ;

(4) $x_{n}(t) = e^{-\frac{t}{n}}$ ;

(5) $x_{n}(t) = \sin t-\sin \frac{t}{n}$ ;

(6) $x_{n}(t) = \frac{n t}{\sqrt{n^{2}+1}}$ ;

(7) $x_{n}(t) = n t e^{-n t}$ ;

(8) $x_{n}(t) = n \sin \frac{t}{n}$ ;

(9) $x_{n}(t) = n \ln \left(1+\frac{t}{n}\right)$ ;

(10) $x_{n}(t) = t^{n}-t^{3 n}$ ;

(11) $x_{n}(t) = \varphi \left(t+\frac{1}{n}\right)$ , where $\varphi \in C(\mathbb {R})$ is a fixed function.

(2)

In the space $C^{1}(\left[0,1\right])$ :

(1) $x_{n}(t) = \frac{t^{n+1}}{n+1}-\frac{t^{n+2}}{n+2}$ ;

(2) $x_{n}(t) = \int_{0}^{t} e^{-\frac{u^{2}}{2 n}} d u$ .

(3)

In the space $l_{p}, 1 \leq p \leq +\infty$ :

(1) $x^{(n)} = (0, \ldots , 0, \underbrace{\frac{1}{n}}_{n}, 0, \ldots )$ ;

(2) $x^{(n)} = (1,0, \ldots , 0, \underbrace{\frac{1}{n}}_{n}, 0, \ldots )$ ;

(3) $x^{(n)} = (\frac{1}{n}, 0, \ldots , 0, \underbrace{\frac{1}{n}}_{n}, 0, \ldots )$ ;

(4) $x^{(n)} = (\underbrace{0, \ldots , 0}_{n-1}, 1, \frac{1}{2}, \frac{1}{3}, \ldots )$ ;

(5) $x^{(n)} = (\underbrace{0, \ldots , 0}_{n-1}, \frac{1}{n}, \frac{1}{n+1}, \ldots )$ ;

(6) $x^{(n)} = \left(1, \frac{1}{2}, \ldots , \frac{1}{n}, 0, \ldots \right)$ ;

(7) $x^{(n)} = \left(1, \frac{1}{\sqrt{2}}, \ldots , \frac{1}{\sqrt{n}}, 0, \ldots \right)$ ;

(8) $x^{(n)} = (\underbrace{\frac{1}{n}, \ldots , \frac{1}{n}}_{n}, 0, \ldots )$ ;

(9) $x^{(n)} = (\underbrace{\frac{1}{\ln n}, \ldots , \frac{1}{\ln n}}_{n}, 0, \ldots )$ ;

(10) $x^{(n)} = (\underbrace{1, \ldots , 1}_{n}, 0, \ldots )$ ;

(11) $x^{(n)} = \left(\frac{1}{n}, \frac{1}{n-1}, \ldots , \frac{1}{2}, 1,0, \ldots \right)$ ;

(12) $x^{(n)} = (\sqrt[n]{1}, \sqrt[n]{2}, \ldots , \sqrt[n]{n}, 0, \ldots )$ ;

(13) $x^{(n)}(\underbrace{\left(\frac{1}{2}\right)^{n},\left(\frac{2}{3}\right)^{n}, \ldots ,\left(\frac{n}{n+1}\right)^{n}}_{n}, 0,0, \ldots ).$

(4)

In the space $L_{p}(\left[0,1\right]), 1 \leq p < +\infty$ :

(1) $x_{n}(t) = t^{n}$ ;

(2) $x_{n}(t) = t^{n}-t^{n+1}$ ;

(3) $x_{n}(t) = n \cdot \chi_{\left[0, \frac{1}{n^{2}}\right]}(t)$ ;

(4) $x_{n}(t) = n e^{-n t}, p = 1$ ;

(5) $x_{n}(t) = 1-\frac{t}{n}, p = 2$ ;

(6) $x_{n}(t) = (\sqrt{n}-n \sqrt{n} t) \chi_{\left[0, \frac{1}{n}\right]}(t)$ , $p = 2$ .

Problem 1.10

(1)(2)(3)

Prove directly that the following spaces are complete:

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(1)

$\mathbb {C}^{m}$ with the norm $\left\| x\right\|_{\infty } = \max_{1 \leq k \leq m}\left|x_{k}\right|$ ;

(2)

$l_{p}, 1 \leq p < +\infty$ , with the norm $\left\| x\right\|_{p} = \left(\sum_{k = 1}^{\infty }\left|x_{k}\right|^{p}\right)^{\frac{1}{p}}$ ;

(3)

$l_{\infty }$ with the norm $\left\| x\right\|_{\infty } = \sup_{k \geq 1}\left|x_{k}\right|$ .

Problem 1.11

Let $X$ be a Banach space. Prove that the space of continuous bounded functions $C_{b}(\mathbb {R}, X)$ with uniform norm $\left\| x\right\| = \sup_{t \in \mathbb {R}}\left\| x(t)\right\|_{X}, x \in C_{b}(\mathbb {R}, X)$ , is a Banach space.

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Problem 1.12

(1)(2)

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(1)

Let $X = \mathbb {R}^{2}$ with the norm $\left\| \cdot \right\|_{p}$ , where $p = 1,2,+\infty$ . Draw unit balls centered at the origin.

(2)

Let $X = \mathbb {R}^{3}$ with the norm $\left\| \cdot \right\|_{p}$ , where $p = 1,2,+\infty$ . Draw unit balls centered at the origin.

Problem 1.13

Prove that $C(\left[a, b\right]) \subset L_{p}(\left[a, b\right])$ for all $1 \leq p \leq +\infty$ and $\left[a, b\right] \subset \mathbb {R}$ ; moreover, $\left\| x\right\|_{L_{p}(\left[a, b\right])} \leq (b-a)^{\frac{1}{p}}\left\| x\right\|_{C(\left[a, b\right])}$ for $1 \leq p < +\infty$ and $\left\| x\right\|_{L_{\infty }(\left[a, b\right])} = \left\| x\right\|_{C(\left[a, b\right])}, x \in C(\left[a, b\right])$ .

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Problem 1.14

(1)(2)(3)

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(1)

Show that every norm on the space $\mathbb {R}$ has form $\left\| x\right\| = \alpha \left|x\right|, x \in \mathbb {R}$ , where $\alpha > 0$ is a fixed constant.

(2)

Let $f: \mathbb {R} \rightarrow (0,+\infty )$ be a $\pi$ -periodic function, for which the set $A_{f} = \left\{ (\rho \cos \varphi , \rho \sin \varphi ) \mid \varphi \in \mathbb {R}, 0 \leq \rho \leq f(\varphi )\right\} \subset \mathbb {R}^{2}$ is convex. Put $\left\| x\right\| = \frac{\rho }{f(\varphi )}$ , where $(\rho , \varphi )$ are the polar coordinates of a point $x \in \mathbb {R}^{2}$ . Prove that $\left\| \cdot \right\|$ is a norm on $\mathbb {R}^{2}$ .

(3)

Prove that each norm on $\mathbb {R}^{2}$ can be obtained in the way described in (2)*. Find the functions $f$ that correspond to norms on the spaces $\mathbb {R}_{p}^{2}, 1 \leq p \leq +\infty$ .

Problem 1.15

Prove that for every $1 \leq p_{1} < p_{2} \leq +\infty$ the inclusion $l_{p_{1}} \subset l_{p_{2}}$ holds, and $\left\| x\right\|_{\infty } = \lim_{p \rightarrow \infty }\left\| x\right\|_{p}$ for all $x \in l_{1}$ .

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Problem 1.16

Let $(T, \mathcal{F}, \mu )$ be a space with a finite measure. Prove that for every $1 \leq p_{1} < p_{2} \leq +\infty$ the inclusion $L_{p_{2}}(T, \mu ) \subset L_{p_{1}}(T, \mu )$ holds, and $\left\| x\right\|_{\infty } = \lim_{p \rightarrow \infty }\left\| x\right\|_{p}$ for all $x \in L_{\infty }(T, \mu )$ .

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Problem 1.17

Prove that the spaces $L_{p_{1}}(\mathbb {R})$ and $L_{p_{2}}(\mathbb {R})$ do not contain each other for any $1 \leq p_{1} < p_{2} \leq +\infty$ .

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Problem 1.18

(1)(2)

Let $(T, \mathcal{F}, \mu )$ be a space with a measure, $1 \leq p_{1} < p_{2} < +\infty$ . Prove that

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(1)

$L_{p_{2}}(T, \mu ) \subset L_{p_{1}}(T, \mu )$ if and only if there exists $C > 0$ such that $A \in \mathcal{F}, \mu (A) < \infty \Rightarrow \mu (A) \leq C$ .

(2)

$L_{p_{1}}(T, \mu ) \subset L_{p_{2}}(T, \mu )$ if and only if there exists $\delta > 0$ such that $A \in \mathcal{F}, \mu (A) > 0 \Rightarrow \mu (A) \geq \delta$ .

Problem 1.19

Let $1 \leq p < r < s$ . Prove that $\left(L_{p}(\mathbb {R}) \cap L_{s}(\mathbb {R})\right) \subset L_{r}(\mathbb {R})$ .

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Problem 1.20

Let $x \in L_{\infty }(T, \mu )$ . Prove that

\begin{aligned} \underset {t \in T}{\operatorname {ess} \sup }\left|x(t)\right| & = \min \left\{ C \geq 0 \mid \left|x(t)\right| \leq C(\bmod \mu )\right\} = \\ & = \inf \left\{ \sup _{t \in T \backslash A}\left|x(t)\right| \mid A \in \mathcal{F}, \mu (A) = 0\right\} . \end{aligned}

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Problem 1.21

(1)(2)

?

(1)

Is the function $x \mapsto \left\| x\right\| = \left(\sum_{i = 1}^{\infty }\left|x_{i}\right|^{p}\right)^{\frac{1}{p}}$ a norm on $l_{p}$ for $0 < p < 1$ , where $l_{p}$ for $0 < p < 1$ is formally defined in the same way as for $1 \leq p < +\infty$ ?

(2)

Is the function $x \mapsto \left\| x\right\| = \left(\int_{a}^{b}\left|x(t)\right|^{p} d t\right)^{\frac{1}{p}}$ a norm on $C(\left[a, b\right])$ for $0 < p <$ 1?

Problem 1.22

Under what condition on a sequence $\left\{ \alpha_{n}: n \geq 1\right\} \subset [0,+\infty )$ the function $l_{2} \ni x \mapsto \left(\sum_{k = 1}^{\infty } \alpha_{k}\left|x_{k}\right|^{2}\right)^{1 / 2}$ is a norm on $l_{2}$ ?

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Problem 1.23

(1)(2)

When is equality achieved

?

(1)

in Hölder's inequality;

(2)

in the Minkowski inequality?

Problem 1.24

Let $f \in L_{2}(\mathbb {R}), t f(t) \in L_{2}(\mathbb {R})$ . Prove that $f \in L_{1}(\mathbb {R})$ .

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Problem 1.25

Prove that for $1 \leq p \leq 2$ and for each function $f \in L_{p}(\mathbb {R})$ there exist $f_{1} \in L_{1}(\mathbb {R})$ and $f_{2} \in L_{2}(\mathbb {R})$ such that $f = f_{1}+f_{2}$ .

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Problem 1.26

Let $f \in L_{1}(\mathbb {R})$ be such that for some $\sigma > 0$

\exists A > 0: \int _{\mathbb {R}}\left|x^{k} f(x)\right| d x \leq A \sigma ^{k}, k \geq 1.

Prove that $f = 0(\bmod m)$ on $(-\infty ,-\sigma ) \cup (\sigma ,+\infty )$ .

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Problem 1.27

Let $0 < \alpha < 1$ . For which $1 \leq p < +\infty$ the function

x(t) = \begin{cases} n^{\alpha }, & \frac{1}{n+1} \leq t < \frac{1}{n}, n \geq 1, \\ 0, & t = 0, t = 1, \end{cases}

belongs to the space $L_{p}(\left[0,1\right])$ ?

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Problem 1.28

(1)(2)

Give examples of infinite-dimensional subspaces $L$ in $L_{1}(\mathbb {R})$ such that

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(1)

$L$ consists of continuous functions;

(2)

$L$ does not contain any nonzero continuous function.

Problem 1.29

Let

X = \left\{ x = \left(x_{1}, x_{2}, \ldots \right)\left|\sup _{n \geq 1}\right| \sum _{k = 1}^{n} x_{k} \mid < +\infty , x_{k} \in \mathbb {K}, k \geq 1\right\}

Define a norm on $X$ as $\left\| x\right\| = \sup_{n \geq 1}\left|\sum_{k = 1}^{n} x_{k}\right|$ . Prove that $(X,\left\| \cdot \right\| )$ is a Banach space.

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Problem 1.30

Prove that the functions $1, \cos t, \cos^{2} t$ are linearly independent in $C(\left[0, \pi \right])$ , and the functions $1, \cos^{2} t, \cos 2 t$ are linearly dependent in $C(\left[0, \pi \right])$ .

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Problem 1.31

(1)(2)

Define the function $\left\| x\right\| = \max_{t \in [a, b]}\left|x^{\prime }(t)-x(t)\right|, x \in C^{1}(\left[a, b\right])$ . Is it a norm

?

(1)

on the set $P_{n}$ of polynomials with powers at most $n$ , where $n \in \mathbb {N}$ is fixed?

(2)

on $C^{1}(\left[a, b\right])$ ?

Problem 1.32

(1)(2)(3)

Let $\left\| \cdot \right\|_{1},\left\| \cdot \right\|_{2}$ be norms on a linear space $X$ . Prove that the following conditions are equivalent:

?

(1)

the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ are equivalent;

(2)

$x_{n} \rightarrow x, n \rightarrow \infty$ , in $(X,\left\| \cdot \right\|_{1})$ if and only if $x_{n} \rightarrow x, n \rightarrow \infty$ , in $(X,\left\| \cdot \right\|_{2})$ ;

(3)

the topologies (i.e. classes of open sets) in the spaces $(X,\left\| \cdot \right\|_{1})$ and $(X,\left\| \cdot \right\|_{2})$ coincide.

Problem 1.33

(1)(2)

Let $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ be equivalent norms on a linear space $X$ . Prove that

?

(1)

if $(X,\left\| \cdot \right\|_{1})$ is a Banach space, then $(X,\left\| \cdot \right\|_{2})$ is also a Banach space;

(2)

if $(X,\left\| \cdot \right\|_{1})$ is a separable space, then the space $(X,\left\| \cdot \right\|_{2})$ is also separable.

Problem 1.34

Let the spaces $(X_{1},\left\| \cdot \right\|_{1})$ and $(X_{2},\left\| \cdot \right\|_{2})$ be isomorphic, and $(X_{1},\left\| \cdot \right\|_{1})$ be a Banach space. Prove that $(X_{2},\left\| \cdot \right\|_{2})$ is also a Banach space. As a corollary, show that every finite-dimensional NLS is a Banach space.

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Problem 1.35

Prove that the norms $\left\| x\right\| = \max_{a \leq t \leq b}\left|x(t)\right|$ and $\left\| x\right\| = \left(\int_{a}^{b}\left|x(t)\right|^{p} d t\right)^{\frac{1}{p}}, 1 \leq p < +\infty$ , in $C(\left[a, b\right])$ are not equivalent.

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Problem 1.36

Is it true that the norms $\left\| x\right\| = \left|x(a)\right|+\max_{a \leq t \leq b}\left|x^{\prime }(t)\right|$ and $\left\| x\right\| = \max_{a \leq t \leq b}\left|x(t)\right|+ \max_{a \leq t \leq b}\left|x^{\prime }(t)\right|$ in $C^{1}(\left[a, b\right])$ are equivalent?

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Problem 1.37

Let $\left\{ r_{n}: n \geq 1\right\} = \mathbb {Q} \cap [0,1]$ . Prove that $\left\| x\right\| = \sum_{n = 1}^{\infty } \frac{\left|x\left(r_{n}\right)\right|}{n^{2}}, x \in C(\left[0,1\right])$ , is a norm on $C(\left[0,1\right])$ . Is it equivalent to the uniform norm?

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Problem 1.38

Let $\left\| \cdot \right\|_{1},\left\| \cdot \right\|_{2}$ be norms on the linear space $X$ , and $\bar{B}_{i}(0,1)$ be the unit closed ball in $\left(X,\left\| \cdot \right\|_{i}\right), i = 1,2$ . Prove that if $\bar{B}_{1}(0,1) = \bar{B}_{2}(0,1)$ , then $\left\| x\right\|_{1} = \left\| x\right\|_{2}, x \in X$ .

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Problem 1.39

(1)(2)(3)

Let $X$ be an NLS, $\left\{ x_{n}: n \geq 1\right\} \subset X$ . Which of the following conditions are equivalent:

?

(1)

$\left\{ x_{n}: n \geq 1\right\}$ is a Cauchy sequence;

(2)

$\left\| x_{n}-x_{m}\right\| \rightarrow 0, m, n \rightarrow \infty$ ;

(3)

$\lim_{m \rightarrow \infty } \lim_{n \rightarrow \infty }\left\| x_{m}-x_{n}\right\| = 0$ ?

Problem 1.40

Prove that an NLS $X$ is a Banach space if and only if any series $\sum_{k = 1}^{\infty } x_{k}$ , for which $\sum_{k = 1}^{\infty }\left\| x_{k}\right\| < +\infty$ , converges in $X$ .

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Problem 1.41

Let $n \geq 1$ . Prove that $C^{n}(\left[a, b\right])$ is a Banach space with the norm

\left\| x\right\| _{n} = \sum _{k = 0}^{n} \max _{a \leq t \leq b}\left|x^{(k)}(t)\right|,

but it is not complete with the norm $\left\| x\right\|_{n-j} = \sum_{k = 0}^{n-j} \max_{a \leq t \leq b}\left|x^{(k)}(t)\right|$ for every $j, 1 \leq j \leq n$ .

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Problem 1.42

(1)(2)(3)

Let $X$ be an NLS, $x \in X$ , and $M \subset X$ be some nonempty set.

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(1)

Prove that the function $\varphi (x) = \rho (x, M)$ is continuous on $X$ .

(2)

Prove that $\rho (x, M) = \rho (x, \bar{M})$ , where $\bar{M}$ is the closure of the set $M$ .

(3)

Prove statements (1) and (2) for the case when $(X, \rho )$ is a metric space and $\rho (x, M): = \inf_{y \in M} \rho (x, y)$ .

Problem 1.43

Let $Q$ be a metric space, and the Banach space $C_{b}(Q)$ of bounded continuous functions on $Q$ with norm $\left\| x\right\| = \sup_{t \in Q}\left|x(t)\right|$ be separable. Prove that $Q$ is a compact set.

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Problem 1.44

(1)(2)

Let $T$ be some set, and $B(T)$ be the space of bounded functions on $T$ with the norm $\left\| x\right\| = \sup_{t \in T}\left|x(t)\right|$ .

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(1)

Prove that $B(T)$ is a Banach space.

(2)

Under what condition on $T$ is the space $B(T)$ separable?

Problem 1.45

Prove that the space $B V_{0}(\left[a, b\right])$ of functions of bounded variation on $\left[a, b\right]$ such that $x(a) = 0$ , with the norm $\left\| x\right\| = V(x,\left[a, b\right])$ is a Banach space. Is it separable?

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Problem 1.46

Let $H^{\lambda }(\left[a, b\right])$ denote the set of all functions that satisfy the Hölder condition on $\left[a, b\right]$ with the exponent $\lambda \in (0,1]$ :

\varphi _{\lambda }(x) = \sup _{a \leq s < t \leq b} \frac{\left|x(t)-x(s)\right|}{(t-s)^{\lambda }} < +\infty .

Prove that $H^{\lambda }(\left[a, b\right])$ is a Banach space with respect to the norm

\left\| x\right\| = \max _{a \leq t \leq b}\left|x(t)\right|+\varphi _{\lambda }(x), x \in H^{\lambda }(\left[a, b\right])

Is it separable?

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Problem 1.47

(1)(2)

Let $X$ be a linear metric space with a metric $\rho$ having the following properties:

(i) $\forall x, y, z \in X: \rho (x+z, y+z) = \rho (x, y)$ ;

(ii) $\forall x \in X \forall \lambda \in \mathbb {C}: \rho (0, \lambda x) = \left|\lambda \right| \rho (0, x)$ .

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(1)

Show that $X$ is an NLS with the norm $\left\| x\right\| = \rho (0, x)$ .

(2)

Give an example of a metric in $\mathbb {R}$ that does not have any of the properties (i), (ii).

Problem 1.48

(1)(2)

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(1)

Let $X = \mathbb {R}^{m}$ with Euclidean norm. Prove that $x$ and $y$ are linearly dependent. Is this correct in other normed spaces?

(2)

Problem 1.49

(1)(2)

Let $X$ be an NLS. Prove that

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(1)

$\forall x, y \in X:\left\| x\right\| \leq \max \left\{ \left\| x+y\right\| ,\left\| x-y\right\| \right\}$ ;

(2)

Problem 1.50

(1)(2)

Let $X$ be the linear space of all complex numerical sequences. Prove that

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(1)

there is no norm on $X$ such that convergence in the space $(X,\left\| \cdot \right\| )$ is equivalent to coordinate-wise convergence;

(2)

there exists a metric on $X$ such that convergence in the space $(X, \rho )$ is equivalent to coordinate-wise convergence.

Problem 1.51

Let $x \in L_{1}(\left[a, b\right])$ . Prove that $\lim_{h \rightarrow 0+} \int_{a}^{b-h}\left|x(t+h)-x(t)\right| d t = 0$ .

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Problem 1.52

Prove that

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(1)

the system of functions $\left\{ 1, t, t^{2}, \ldots \right\}$ is total in $C(\left[a, b\right])$ and in $L_{p}(\left[a, b\right]), 1 \leq p < +\infty$ ;

(2)

the system of functions $\left\{ t^{2 k}: k \geq 0\right\}$ is total in $C(\left[a, b\right])$ and in $L_{p}(\left[a, b\right]), 1 \leq p < +\infty$ , for $a \geq 0$ ;

(3)

the system of functions $\left\{ 1, \cos n t, \sin n t: n \geq 1\right\}$ is total in $L_{p}(\left[0,2 \pi \right]), 1 \leq p < +\infty$ ;

(4)

the system of functions $\left\{ e^{i n t}: n \in \mathbb {Z}\right\}$ is total in $L_{p}(\left[0,2 \pi \right]), 1 \leq p < +\infty$ ;

(5)

the system of functions $\left\{ e^{i n t}: n \in \mathbb {Z}\right\}$ is total in $C(\left[a, b\right])$ if and only if $\left|b-a\right| < 2 \pi$ ;

(6)

the system of functions $\left\{ 1, \sin n t: n \geq 1\right\}$ is total in $C(\left[0, b\right])$ if and only if $0 < b < \pi$ ;

(7)

the system of functions $\left\{ 1, \cos n t: n \geq 1\right\}$ is total in $C(\left[0, b\right])$ if and only if $0 < b \leq \pi ;$

(8)

the system of elements $\left\{ e_{n} = (0, \ldots , 0, \underbrace{1}_{n}, 0, \ldots ): n \geq 1\right\}$ is total in $l_{p}, 1 \leq p < +\infty$ .

Problem 1.53

Prove that the following sets are dense in $L_{p}(\left[a, b\right]), 1 \leq p < +\infty$ :

?

(1)

simple functions;

(2)

step functions;

(3)

continuous functions;

(4)

polynomials;

(5)

polynomials with rational coefficients;

(6)

polynomials with zero sum of coefficients;

(7)

even polynomials if $a \geq 0$ ;

(8)

continuous functions $x$ such that $x(a) = 0$ ;

(9)

continuous functions $x$ such that $x(a) = x(b) = 0$ ;

(10)

polynomials of $e^{t}$ .

Problem 1.54

(1)(2)

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(1)

Prove that the set $C_{0}(\mathbb {R})$ of continuous functions on $\mathbb {R}$ which vanish outside some finite interval is dense in $L_{p}(\mathbb {R}), 1 \leq p < +\infty$ .

(2)

Prove that the space $L_{p}(\mathbb {R}), 1 \leq p < +\infty$ is separable.

Problem 1.55

Prove that the set $C_{b}(\mathbb {R})$ of bounded continuous functions on $\mathbb {R}$ is not dense in $L_{\infty }(\mathbb {R})$ .

?

Problem 1.56

Prove the nonseparability of the space $L_{\infty }(\left[a, b\right])$ .

?

Problem 1.57

Let $K$ be a compact set in the metric space $(\mathbb {R}^{m}, \rho )$ . Prove that the space $C(K)$ with uniform norm is separable.

?

Problem 1.58

(1)(2)(3)

Let $(K, \rho )$ be a compact metric space. Prove that

?

(1)

if $K$ is an infinite set, then $C(K)$ is an infinite-dimensional space;

(2)

if $\left\{ U_{i}: 1 \leq i \leq n\right\}$ is a finite open cover of $K$ , then there exists a system of real continuous functions $\left\{ \varphi_{i}: 1 \leq i \leq n\right\}$ on $K$ such that

(i) $\varphi_{i}(t) \geq 0,1 \leq i \leq n, t \in K$ ,

(ii) $\varphi_{i}(t) = 0, t \notin U_{i}, i = 1, \ldots , n$ ,

(iii) $\sum_{i = 1}^{n} \varphi_{i}(t) = 1, t \in K$ (such a system is called a partition of unity);

(3)

the Banach space $C(K)$ is separable.

Problem 1.59

Let $X$ be an NLS. Prove that $X$ is separable if and only if there exists $r > 0$ such that the sphere $S(0, r)$ is a separable metric space.

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Problem 1.60

Let $p \in [1, \infty )$ . The space of locally $p$ -integrable functions on $\mathbb {R}$ is defined as

L_{p}^{l o c}(\mathbb {R}): = \left\{ f: \mathbb {R} \rightarrow \mathbb {C} \mid \forall a, b \in \mathbb {R}, a < b: f \cdot \chi _{\left[a, b\right]} \in L_{p}(\mathbb {R})\right\} .

Under what condition on a function $g \in L_{p}^{\text{loc }}(\mathbb {R})$ the set

\left\{ g \cdot \chi _{(\alpha , \beta ]} \mid -\infty < \alpha < \beta < +\infty \right\}

is total in $L_{p}(\mathbb {R})$ ?

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Problem 1.61

Are the following subsets subspaces in $C(\left[-1,1\right])$ :

?

(1)

monotonic functions;

(2)

nondecreasing functions;

(3)

even functions;

(4)

odd functions;

(5)

polynomials;

(6)

polynomials of degree at most $m$ , where $m \in \mathbb {N}$ is fixed;

(7)

continuous piecewise smooth functions;

(8)

continuously differentiable functions;

(9)

continuous functions of bounded variation;

(10)

functions $x$ for which $x(0) = 0$ ;

(11)

functions $x$ for which $\int_{-1}^{1} x(t) d t = 0$ ;

(12)

functions satisfying the Lipschitz condition.

Remark.

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Piecewise smooth functions are the functions that have a continuous derivative everywhere except for a finite number of points.

Problem 1.62

(1)(2)

Is the set

M = \left\{ x \in C^{1}(\left[-1,1\right]) \mid \int _{-1}^{1} x(t) d t = 0\right\}

a subspace

?

(1)

in the space $C^{1}(\left[-1,1\right])$ with the norm

\left\| x\right\| = \max _{-1 \leq t \leq 1}\left|x(t)\right|+\max _{-1 \leq t \leq 1}\left|x^{\prime }(t)\right|?

(2)

in the space $C(\left[-1,1\right])$ with the norm $\left\| x\right\| = \max_{-1 \leq t \leq 1}\left|x(t)\right|$ ?

Problem 1.63

Consider $l_{1}$ as a subset of $l_{\infty }$ . Find its closure in $l_{\infty }$ .

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Problem 1.64

(1)(2)(3)

Let $M = \left\{ x = \left(x_{1}, x_{2}, \ldots \right) \in X \mid \sum_{k = 1}^{\infty } x_{k} = 0\right\}$ . Is $M$ a subspace of $X$ if

?

(1)

$X = l_{1}$ ;

(2)

$X = l_{p}, 1 < p < +\infty$ ;

(3)

$X = l_{\infty }$ ?

Problem 1.65

Is the space $l_{1}$ complete with the norm of the space $l_{2}$ ?

?

Problem 1.66

(1)(2)

Let us define in $l_{\infty }$ the set $c = \left\{ x \in l_{\infty } \mid \exists \lim_{i \rightarrow \infty } x_{i} \in \mathbb {K}\right\}$ . Prove that

?

(1)

$c$ is a subspace in $l_{\infty }$ ;

(2)

$c$ with the norm $\left\| x\right\| = \left\| x\right\|_{\infty } = \sup_{i \geq 1}\left|x_{i}\right|$ is a separable Banach space.

Problem 1.67

(1)(2)

Let us define in $l_{\infty }$ the set $c_{0} = \left\{ x \in l_{\infty } \mid \lim_{k \rightarrow \infty } x_{k} = 0\right\}$ . Prove that

?

(1)

$c_{0}$ is a subspace in $l_{\infty }$ ;

(2)

$c_{0}$ with the norm $\left\| x\right\| = \left\| x\right\|_{\infty } = \sup_{i \geq 1}\left|x_{i}\right|$ is a separable Banach space.

Problem 1.68

Let $A$ be a set in NLS. Is it always true that $\operatorname {span}(\bar{A}) = \overline{\operatorname {span}(A)}$ ?

?

Problem 1.69

Prove that a subspace in NLS is a convex set.

?

Problem 1.70

Prove that the intersection of any family of convex sets in a linear space is a convex set.

?

Problem 1.71

Let a function $\varphi$ on a linear space $X$ satisfy the first two axioms of the norm. Prove that $\varphi$ is a norm on $X$ if and only if the set $B = \left\{ x \in X \mid \varphi (x) \leq 1\right\}$ is convex.

?

Problem 1.72

Prove that in any finite-dimensional NLS all norms are equivalent.

?

Problem 1.73

Prove that in any finite-dimensional NLS every bounded closed set is compact.

?

Problem 1.74

(1)(2)

?

(1)

Prove that in a finite-dimensional NLS every bounded sequence has at least one partial limit (i.e. the limit of some subsequence).

(2)

Give an example of a bounded sequence in $l_{2}$ which does not have any partial limits.

Problem 1.75

Prove that in every NLS $X$ the distance from any fixed point $x$ to any finite-dimensional subspace $M$ is attained.

?

Problem 1.76

Let $X$ be an NLS, $x_{1}, \ldots , x_{n} \in X$ . Prove that

\operatorname {span}\left(\left\{ x_{1}, \ldots , x_{n}\right\} \right) = \overline{\operatorname {span}\left(\left\{ x_{1}, \ldots , x_{n}\right\} \right)}

?

Problem 1.77

Let $X$ be an NLS, $x_{1}, \ldots , x_{n} \in X$ be linearly independent elements. Prove that $\rho \left(x_{n}, \operatorname {span}\left(\left\{ x_{1}, \ldots , x_{n-1}\right\} \right)\right) > 0$ .

?

Problem 1.78

Let $L_{1}, L_{2}$ be subspaces in NLS $X$ , and at least one of these subspaces is finite-dimensional. Prove that $L_{1}+L_{2}$ is a subspace in $X$ .

?

Problem 1.79

(1)(2)

Let $(X_{1},\left\| \cdot \right\|_{X_{1}})$ and $(X_{2},\left\| \cdot \right\|_{X_{2}})$ be NLS over a field $\mathbb {K}$ . The Cartesian product $X: = X_{1} \times X_{2}$ , in which linear operations are defined as follows:

\left(x_{1}, x_{2}\right)+\left(y_{1}, y_{2}\right) = \left(x_{1}+y_{1}, x_{2}+y_{2}\right), \quad \alpha \left(x_{1}, x_{2}\right) = \left(\alpha x_{1}, \alpha x_{2}\right),

$\left(x_{1}, x_{2}\right) \in X,\left(y_{1}, y_{2}\right) \in X, \alpha \in \mathbb {K}$ , is called the direct sum of $X_{1}$ and $X_{2}$ . It is denoted by $X_{1} \bigoplus X_{2} = X$ .

?

(1)

Prove that $X$ is a linear space.

(2)

Prove that the functions

\begin{gathered} \left\| \left(x_{1}, x_{2}\right)\right\| _{p} = \left(\left\| x_{1}\right\| _{X_{1}}^{p}+\left\| x_{2}\right\| _{X_{2}}^{p}\right)^{\frac{1}{p}}, 1 \leq p < +\infty \\ \left\| \left(x_{1}, x_{2}\right)\right\| _{\infty } = \max \left\{ \left\| x_{1}\right\| _{X_{1}},\left\| x_{2}\right\| _{X_{2}}\right\} \end{gathered}

are norms in $X$ .

Problem 1.80

Let $X$ be an NLS over a field $\mathbb {K}$ . Prove that a set $M \subset X$ is bounded if and only if for each sequence $\left\{ x_{n}: n \geq 1\right\} \subset M$ and each sequence $\left\{ \alpha_{n}: n \geq 1\right\} \subset \mathbb {K}$ such that $\alpha_{n} \rightarrow 0, n \rightarrow \infty$ , the convergence $\alpha_{n} x_{n} \rightarrow 0, n \rightarrow \infty$ , in $X$ holds.

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Problem 1.81

Let $X$ be an NLS, and $M \subset X$ be a linear set, $M \neq X$ . Prove that $M$ has no interior points.

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Problem 1.82

(1)(2)(3)(4)

A linearly independent system $\left\{ x_{\alpha }: \alpha \in A\right\}$ of elements of a linear space $X$ is called a Hamel basis if $\operatorname {span}\left(\left\{ x_{\alpha }: \alpha \in A\right\} \right) = X$ .

?

(1)

Prove that in every linear space there exists a Hamel basis.

(2)

Prove that each element of the space $X$ is uniquely represented as a linear combination of a finite number of elements from the Hamel basis of the space $X$ .

(3)

Based on the Baire category theorem (see, for example, [26], Theorem III. 8 or [21], Ch. 2, Sec. 7, Theorem 2), prove that in an infinite-dimensional Banach space, there is no countable Hamel basis.

(4)

Show that in item (3)* completeness of the space is essential.

Problem 1.83

Prove that an NLS is nonseparable if and only if it contains an uncountable number of balls of some fixed radius $r > 0$ that do not intersect pairwise.

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Problem 1.84

Let $X$ be a Banach space, $\left\{ \bar{B}\left(x_{n}, r_{n}\right): n \geq 1\right\}$ be a sequence of nested closed balls, and $\lim_{n \rightarrow \infty } r_{n} = 0$ . Prove that

\exists !x \in \bigcap _{n = 1}^{\infty } \bar{B}\left(x_{n}, r_{n}\right)

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Problem 1.85

Let $X$ be an NLS in which any sequence $\left\{ \bar{B}\left(x_{n}, r_{n}\right): n \geq 1\right\}$ of nested closed balls, for which $\lim_{n \rightarrow \infty } r_{n} = 0$ , has a nonempty intersection. Prove that $X$ is a Banach space.

?

Problem 1.86

(1)(2)

?

(1)

Give an example of a Banach space $X$ and a sequence of nested nonempty closed sets that has an empty intersection.

(2)

Prove that every sequence of nested closed balls in a Banach space has a nonempty intersection.

Problem 1.87

Let $X$ be a Banach space, $M \subset X$ be a linear set. Prove that the completion of $M$ in the norm of the space $X$ coincides with the closure of $M$ .

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Problem 1.88

Let $X$ be an NLS. Consider the set $\bar{X}$ , which consists of all Cauchy sequences of the space $X$ , i.e. $\bar{x} \in \bar{X}$ , if $\bar{x} = \left(x_{1}, x_{2}, \ldots \right),\left\{ x_{n}: n \geq 1\right\}$ is a Cauchy sequence in $X$ .

?

(1)

Two sequences $\bar{x} = \left\{ x_{n}: n \geq 1\right\}$ and $\bar{y} = \left\{ y_{n}: n \geq 1\right\}$ are said to be equivalent $(\bar{x} \sim \bar{y})$ , if $\left\| x_{n}-y_{n}\right\| \rightarrow 0, n \rightarrow \infty$ . Prove that the relation $\bar{x} \sim \bar{y}$ is an equivalence relation, i.e. it is reflexive, symmetric, and transitive.

(2)

Thus the set $\bar{X}$ is split into classes of equivalent sequences. Let us denote the set of these classes by $\widetilde{X}$ . Prove that it is possible to introduce the structure of a linear space in $\widetilde{X}$ .

(3)

For every $\tilde{x} \in \tilde{X}$ , we set $\left\| \tilde{x}\right\|_{\tilde{X}} = \lim_{n \rightarrow \infty }\left\| x_{n}\right\|_{X}$ . Prove that this formula determines a norm on $\widetilde{X}$ .

(4)

Prove that $\left(\widetilde{X},\left\| \tilde{x}\right\|_{\tilde{X}}\right)$ is a complete space.

(5)

Prove that $\underset {\sim }{X}$ is dense in $\widetilde{X}$ if an element $x \in X$ is identified with the equivalence class $\tilde{x} \in \tilde{X}$ that contains the sequence $(x, x, \ldots )$ .

Problem 1.89

(1)(2)

Let us call the sequence $\left\{ e_{1}, \ldots , e_{n}, \ldots \right\} \subset X$ a Schauder basis in the NLS $X$ if for each $x \in X$ there exists a unique representation $x = \sum_{n = 1}^{\infty } \alpha_{n} e_{n}, \alpha_{n} \in \mathbb {K}, n \geq$ 1, where the series converges in norm.

?

(1)

Prove that any space with a Schauder basis is separable.

(2)

Prove that there exists a Schauder basis in the spaces $l_{p}, 1 \leq p < +\infty , c_{0}, c$ .

Problem 1.90

Let $\left\{ e_{n}: n \geq 1\right\} \subset X$ be the sequence of nonzero elements in a Banach space $X$ with the following properties:

(a) $\overline{\operatorname {span}\left(\left\{ e_{n}: n \geq 1\right\} \right)} = X$ ;

(b) $\exists C > 0 \forall m > k \forall \lambda_{1}, \ldots , \lambda_{m} \in \mathbb {K}:\left\| \sum_{i = 1}^{k} \lambda_{i} e_{i}\right\| \leq C\left\| \sum_{i = 1}^{m} \lambda_{i} e_{i}\right\|$ .

Prove that $\left\{ e_{n}: n \geq 1\right\}$ is a Schauder basis in the space $X$ .

?

Remark.

?

The sequence $\left\{ e_{n}: n \geq 1\right\} \subset X \backslash \left\{ 0\right\}$ is a Schauder basis in the Banach space $X$ if and only if it satisfies conditions (a) and (b) (see problem 8.30*).

Problem 1.91

(1)(2)(3)

Consider the sequence of functions on $\left[0,1\right]$ defined as follows:

h_{2^{k}+l}(t) = \begin{cases} 1, & \frac{2 l-2}{2^{k+1}} \leq t < \frac{2 l-1}{2^{k+1}} \\ -1, & \frac{2 l-1}{2^{k+1}} \leq t < \frac{2 l}{2^{k+1}} \\ 0, & \text{ otherwise } \end{cases}

where $k = 0,1,2, \ldots$ and $1 \leq l \leq 2^{k}, h_{1}(t) = 1, t \in [0,1]$ . The sequence $\left\{ h_{n}: n \geq 1\right\}$ is called the Haar system. Prove that

?

(1)

$\overline{\operatorname {span}\left(\left\{ h_{n}: n \geq 1\right\} \right)} = L_{p}(\left[0,1\right])$ in the norm of $L_{p}(\left[0,1\right]), 1 \leq p < +\infty$ ;

(2)

the Haar system is a Schauder basis in $L_{p}(\left[0,1\right])$ for every $1 \leq p < +\infty$ ;

(3)

$\forall n \geq 1 \forall \lambda_{1}, \lambda_{2}, \ldots , \lambda_{n+1} \in \mathbb {K}:\left\| \sum_{i = 1}^{n} \lambda_{i} h_{i}\right\|_{p} \leq \left\| \sum_{i = 1}^{n+1} \lambda_{i} h_{i}\right\|_{p}$ .

A basis satisfying this condition is said to be monotone.

Problem 1.92

Let $\left\{ h_{n}: n \geq 1\right\}$ be the Haar system from the previous problem. Consider the sequence of functions $\left\{ s_{n}: n \geq 0\right\}$ in $C(\left[0,1\right])$ defined as follows: $s_{0}(t) = 1, t \in \left[0,1\right]$ , and $s_{n}(t) = \int_{0}^{t} h_{n}(\tau ) d \tau , t \in [0,1], n \geq 1$ . The sequence $\left\{ s_{n}: n \geq 0\right\}$ is called the Schauder system. Prove that the Schauder system is a monotone Schauder basis in $C(\left[0,1\right])$ .

?

Problem 1.93

Let $A$ be a set in an NLS $X$ , and let $A$ contains a countable subset which is dense in $A$ . Prove that $\overline{\operatorname {span}(A)}$ is a separable subspace in $X$ .

?

Problem 1.94

(1)(2)

An NLS $X$ is said to be strictly normed if for any $x_{1}, x_{2} \in X$ equalities $\left\| x_{1}\right\| = \left\| x_{2}\right\| = \left\| \frac{1}{2}\left(x_{1}+x_{2}\right)\right\| = 1$ imply that $x_{1} = x_{2}$ . The norm on such a space is said to be strictly convex. Prove that

?

(1)

$X$ is strictly normed if and only if for all $x_{1}, x_{2} \in X$ and $\alpha \in [0,1]$ equalities $\left\| x_{1}\right\| = \left\| x_{2}\right\| = \left\| \alpha x_{1}+(1-\alpha ) x_{2}\right\| = 1$ imply that $x_{1} = x_{2}$ ;

(2)

$X$ is strictly normed if and only if the unit ball $\bar{B}(0,1)$ is a strictly convex set (for an open set $A$ , the set $\bar{A}$ is said to be strictly convex if $\alpha x_{1}+(1-\alpha ) x_{2} \in A$ for all $x_{1}, x_{2} \in \bar{A}$ and $\alpha \in (0,1)$ ).

Problem 1.95

For which $1 \leq p \leq +\infty$ the space $\mathbb {R}^{m}$ with the norm $\left\| \cdot \right\|_{p}$ is strictly normed?

?

Problem 1.96

(1)(2)

Prove that

?

(1)

the spaces $L_{p}(T, \mu ), 1 < p < +\infty$ , are strictly normed;

(2)

the spaces $C(\left[a, b\right]), L_{1}(\left[a, b\right]), L_{\infty }(\left[a, b\right])$ are not strictly normed.

Problem 1.97

(1)(2)

Prove that a real NLS $X$ is strictly normed if and only if for an arbitrary convex set $L \subset X$ and for any $x \in X$ there exists at most one element of the best approximation (i.e. if the element of the best approximation exists, then it is unique).

In $(\mathbb {R}^{2},\left\| \cdot \right\|_{1})$ , give examples of the subspace $L \subset \mathbb {R}^{2}$ and the element $x \in \mathbb {R}^{2}$ , for which

?

(1)

there exists a unique element of the best approximation in $L$ ;

(2)

there exist several elements of the best approximation in $L$ .

Problem 1.98

(1)(2)

A normed linear space $X$ is said to be uniformly convex if

\begin{aligned} \forall \varepsilon > 0 \exists \delta > 0 \forall x_{1}, x_{2} \in X,\left\| x_{1}\right\| & = \left\| x_{2}\right\| = 1, \\ & \left\| \frac{1}{2}\left(x_{1}+x_{2}\right)\right\| > 1-\delta :\left\| x_{1}-x_{2}\right\| < \varepsilon . \end{aligned}

Prove that

?

(1)

a uniformly convex space is strictly normed;

(2)

a finite-dimensional strictly normed space is uniformly convex.

Problem 1.99

(1)(2)

?

(1)

Let $1 < p < \infty$ . Prove that for any $0 < c < 1$ there exists $a_{c} \geq 1$ such that

\frac{\left|z_{1}\right|^{p}+\left|z_{2}\right|^{p}}{2} \leq a_{c}\left(\frac{\left|z_{1}\right|^{p}+\left|z_{2}\right|^{p}}{2}-\left|\frac{z_{1}+z_{2}}{2}\right|^{p}\right)

for all $z_{1}, z_{2} \in \mathbb {K}$ , for which $\left|\frac{z_{1}-z_{2}}{2}\right|^{p} \geq c\left(\frac{\left|z_{1}\right|^{p}+\left|z_{2}\right|^{p}}{2}\right)$ .

(2)

Prove that the space $L_{p}(T, \mu ), 1 < p < +\infty$ , is uniformly convex.

Problem 1.100

Let $X$ be a uniformly convex Banach space. Prove that for every element $x \in X$ in an arbitrary convex closed set $L \subset X$ there exists a unique element of the best approximation.

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Problem 1.101

Let $X$ be a uniformly convex Banach space, $K \subset X$ be a nonempty closed convex set. Prove that the function $f(x) = \left\| x\right\| , x \in X$ , attains its minimum on $K$ exactly once.

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Problem 1.102

Let $(T, \mathcal{F}, \mu )$ be a measurable space with a measure $\mu , \mu (T) < +\infty$ . Let $\Phi$ denote the set of all $n \in \mathbb {N}$ for which there exists a partition of the space $T = \bigcup_{k = 1}^{n} A_{k}$ , where $A_{i} \in \mathcal{F}, \mu \left(A_{i}\right) > 0,1 \leq i \leq n, A_{i} \cap A_{j} = \varnothing , i \neq j$ . Prove that for every $1 \leq p < +\infty$ the dimension of the space $L_{p}(T, \mu )$ is given by $\operatorname {dim} L_{p}(T, \mu ) = \sup \Phi$ . At the same time,

\operatorname {dim} L_{p}(T, \mu ) = n < +\infty \Leftrightarrow \text{ there exists a partition of } T \text{ into } n \text{ atoms. }

(A set $A \in \mathcal{F}$ is called an atom if $\mu (A) > 0$ and for every subset $B \subset A, B \in \mathcal{F}$ , either $\mu (B) = 0$ , or $\mu (A \backslash B) = 0$ .)

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