Math Problems Vault

Problem 3.5

Let $X$ be an NLS over the field $\mathbb {K}$ . Prove that the set of values of a nonzero linear functional on $X$ coincides with $\mathbb {K}$ .

?

Problem 3.6

Let $X$ be an NLS, $f, g \in X^{*}$ . Prove that for any $\alpha , \beta \in \mathbb {K}$ the functional $\alpha f+\beta g$ , where $(\alpha f+\beta g)(x): = \alpha f(x)+\beta g(x), x \in X$ , also belongs to $X^{*}$ .

?

Problem 3.7

Let $X$ be an NLS, $f \in X^{*}$ . Prove that

?

(1)

$\left\| f\right\| = \sup_{\left\| x\right\| \leq 1}\left|f(x)\right|$ ;

(2)

$\left\| f\right\| = \sup_{\left\| x\right\| = 1}\left|f(x)\right|$ ;

(3)

$\left\| f\right\| = \min \left\{ C \geq 0 \mid \left|f(x)\right| \leq C\left\| x\right\| , x \in X\right\}$ ;

(4)

$\left\| f\right\| = \sup_{\left\| x\right\| \leq 1} f(x)$ , if the space $X$ is real;

(5)

$\left\| f\right\| = \sup_{x \in M, x \neq 0} \frac{\left|f(x)\right|}{\left\| x\right\| }$ , where $M$ is a dense subset of $X$ .

Problem 3.8

Let $X$ be an NLS, $f_{n} \in X^{*}$ and $\left\| f_{n}\right\| \leq 1, n \geq 1$ . Set

f(x) = \sum _{n = 1}^{\infty } \frac{1}{n^{2}} f_{n}(x), x \in X

Prove that $f \in X^{*}$ .

?

Problem 3.9

Determine whether the given functionals on $C(\left[0,1\right])$ are linear and continuous. For continuous linear functionals, calculate their norms using the definition:

?

(1)

$f(x) = x(0)$ ;

(2)

$f(x) = \int_{0}^{1} x(t) d t$ ;

(3)

$f(x) = \int_{0}^{1}\left|x(t)\right| d t$ ;

(4)

$f(x) = \left\| x\right\|$ ;

(5)

$f(x) = \sum_{n = 1}^{\infty } \frac{1}{2^{n}} x\left(\frac{1}{2^{n}}\right)$ ;

(6)

$f(x) = \int_{0}^{1} t x(t) d t$ ;

(7)

$f(x) = \int_{0}^{1} x(t) \operatorname {sign}\left(t-\frac{1}{2}\right) d t$ ;

(8)

$f(x) = \int_{0}^{1} \operatorname {sign} x(t) d t$ ;

(9)

$f(x) = \int_{0}^{1} t x^{2}(t) d t$ ;

(10)

$f(x) = \int_{0}^{1}(1-2 t) x(t) d t ;$

(11)

$f(x) = \int_{0}^{1} x(t) \cos \pi t d t$ ;

(12)

$f(x) = \int_{0}^{1} x(t) \sin \pi t d t$ ;

(13)

$f(x) = \int_{0}^{1}\left|x(t)\right| \cos \pi t d t$ ;

(14)

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

(15)

$f(x) = \int_{0}^{\frac{1}{3}} x(t) d t-\int_{\frac{1}{3}}^{1} x(t) d t ;$

(16)

$f(x) = \sum_{k = 1}^{n} \lambda_{k} x\left(t_{k}\right)$ , where $t_{1}, \ldots , t_{n}$ is a set of different points from $\left[0,1\right], \lambda_{1}, \ldots , \lambda_{n}$ is a set of real numbers.

Problem 3.10

Determine whether the given functionals are linear and continuous. For continuous linear functionals, calculate their norms using the definition:

?

(1)

$X = l_{1}, f(x) = \sum_{n = 1}^{\infty } x_{n}$ ;

(2)

$X = l_{1}, f(x) = \sum_{n = 1}^{\infty }\left(1-\frac{1}{n}\right) x_{n}$ ;

(3)

$X = l_{1}, f(x) = \sum_{n = 1}^{\infty } x_{n} y_{n}$ , where $y \in l_{\infty }$ is fixed;

(4)

$X = l_{2}, f(x) = \sum_{n = 1}^{\infty }\left|x_{n}\right|^{2}$ ;

(5)

$X = l_{2}, f(x) = \sum_{n = 1}^{\infty } \frac{x_{n}}{\sqrt{n(n+1)}}$ ;

(6)

$X = l_{2}, f(x) = \sum_{n = 1}^{\infty }(-1)^{n} \frac{x_{n}}{n}$ ;

(7)

$X = l_{\frac{4}{3}}, f(x) = \sum_{n = 1}^{\infty } \frac{x_{2 n}}{3^{n / 4}}$ ;

(8)

$X = l_{1}, f(x) = \sum_{n = 1}^{\infty }\left(1-(-1)^{n}\right) \frac{n-1}{n} x_{n}$ ;

(9)

$X = l_{p}, 1 \leq p < +\infty , f(x) = \sum_{n = 1}^{\infty } \frac{x_{n}}{n}$ ;

(10)

$X = l_{p}, 1 \leq p \leq +\infty , f(x) = \sum_{n = 1}^{\infty } \frac{x_{n}}{2^{n}}$ ;

(11)

$X = l_{p}, 1 \leq p \leq +\infty , f(x) = \operatorname {sign}\left(x_{1}\right)$ ;

(12)

$X = l_{p}, 1 \leq p < +\infty , f(x) = \sum_{n: x_{n} \neq 0} e^{-\frac{1}{\left|x_{n}\right|}}$ ;

(13)

$X = l_{p}, 1 \leq p \leq +\infty , f(x) = x_{j}, j \in \mathbb {N}$ is fixed;

(14)

$X = l_{p}, 1 \leq p \leq +\infty , f(x) = x_{j}+x_{j+1}, j \in \mathbb {N}$ is fixed;

(15)

$X = l_{p}, 1 \leq p \leq +\infty , f(x) = x_{1}+2 x_{2}$ .

Problem 3.11

Determine whether the given functionals are linear and continuous. For continuous linear functionals, calculate their norms using the definition:

?

(1)

$X = L_{2}(\left[0,1\right]), f(x) = \int_{0}^{1} t x(t) d t ;$

(2)

$X = L_{2}(\left[0,1\right]), f(x) = \int_{0}^{1} x(t) \operatorname {sign}\left(t-\frac{1}{2}\right) d t$ ;

(3)

$X = L_{2}(\left[0,1\right]), f(x) = \int_{0}^{1} \frac{x(t)}{\sqrt[4]{t}} d t$ ;

(4)

$X = L_{2}(\left[0,1\right]), f(x) = \int_{0}^{1}\left|x(t)\right|^{\frac{3}{4}} d t$ ;

(5)

$X = L_{p}(\left[0,1\right]), 1 \leq p \leq +\infty , f(x) = \int_{0}^{1} t x(t) d t ;$

(6)

$X = L_{p}\left(\left[0, \frac{\pi }{2}\right]\right), 1 \leq p \leq +\infty , f(x) = \int_{0}^{\frac{\pi }{2}} x(t) \sin t d t ;$

(7)

$X = L_{p}(\left[0,1\right]), 1 \leq p \leq +\infty , f(x) = \int_{0}^{\frac{1}{2}} t x(t) d t ;$

(8)

$X = L_{p}(\left[0,2\right]), 1 \leq p \leq +\infty , f(x) = \int_{0}^{1} x(t) d t-3 \int_{1}^{2} x(t) d t$ ;

(9)

$X = L_{1}(\mathbb {R}), f(x) = \int_{\mathbb {R}} x(t) \operatorname {arctg} t d t$ .

Problem 3.12

Determine whether the given functionals are linear and continuous. For continuous linear functionals, calculate their norms using the definition:

?

(1)

$X = C^{1}(\left[0,1\right])$ with the norm $\left\| x\right\| = \max_{t \in [0,1]}\left|x(t)\right|+\max_{t \in [0,1]}\left|x^{\prime }(t)\right|, f(x) = x^{\prime }(0) ;$

(2)

$X = C^{1}(\left[0,1\right])$ with the uniform norm, $f(x) = x^{\prime }(0)$ ;

(3)

$X = C^{1}(\left[0,1\right])$ with the uniform norm, $f(x) = \int_{0}^{1} x^{\prime }(t) \sin \pi t d t$ ;

(4)

$X = C^{1}(\left[0,1\right])$ with norm $\left\| x\right\| = \max_{t \in [0,1]}\left|x(t)\right|+\max_{t \in [0,1]}\left|x^{\prime }(t)\right|, f(x) = x^{\prime }(0)+ x(1) ;$

(5)

$X = C(\left[0,1\right]), f(x) = \int_{0}^{1} x(\sqrt{t}) d t$ ;

(6)

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

(7)

$X = \left\{ x \in L_{2}(\left[0,1\right])\left|\int_{0}^{1}\right| x\left(t^{\alpha }\right) \mid d t < +\infty \right\} , \alpha > 0,\left\| \cdot \right\|_{X} = \left\| \cdot \right\|_{L_{2}(\left[0,1\right])}$ ,

f(x) = \int _{0}^{1} x\left(t^{\alpha }\right) d t

(8)

$X = C(\left[0,1\right]), f(x) = \lim_{n \rightarrow \infty } \int_{0}^{1} x\left(t^{n}\right) d t$ ;

(9)

$X = c, f(x) = \lim_{n \rightarrow \infty } x_{n}$ ;

(10)

$X = \left\{ x \in l_{2} \mid \sum_{n = 1}^{\infty } x_{n} \sin n \text{ converges}\right\} , \left\| \cdot \right\|_{X} = \left\| \cdot \right\|_{l_{2}}, f(x) = \sum_{n = 1}^{\infty } x_{n} \sin n$ ;

(11)

$X = \left\{ x \in l_{2} \mid \sum_{n = 1}^{\infty } x_{n^{2}} \text{ converges}\right\} , \left\| \cdot \right\|_{X} = \left\| \cdot \right\|_{l_{2}}, f(x) = \sum_{n = 1}^{\infty } x_{n^{2}}$ ;

(12)

$X = \left\{ x \in l_{\infty } \mid \sum_{n = 1}^{\infty } x_{n} y_{n} \text{ converges}\right\}$ , where $\left\{ y_{n}: n \geq 1\right\}$ is a fixed sequence such that the series $\sum_{n = 1}^{\infty } y_{n}$ converges conditionally, $\left\| \cdot \right\|_{X} = \left\| \cdot \right\|_{l_{\infty }}, f(x) = \sum_{n = 1}^{\infty } x_{n} y_{n}$ .

Problem 3.13

Prove that $l_{1} \subset l_{\infty }^{*}$ , i.e. for every element $a = \left(a_{1}, \ldots , a_{n}, \ldots \right) \in l_{1}$ the functional

f_{a}(x) = \sum _{n = 1}^{\infty } a_{n} x_{n}, x = \left(x_{1}, \ldots , x_{n}, \ldots \right) \in l_{\infty }

is linear and continuous on $l_{\infty }$ , and $\left\| f_{a}\right\| = \left\| a\right\|_{1}$ .

?

Problem 3.14

Prove that $L_{1}(T, \mu ) \subset \left(L_{\infty }(T, \mu )\right)^{*}$ , i.e. for every element $h \in L_{1}(T, \mu )$ the functional $f(x) = \int_{T} h(t) x(t) d \mu (t), x \in L_{\infty }(T, \mu )$ , is linear and continuous on $L_{\infty }(T, \mu )$ , and $\left\| f\right\| = \left\| h\right\|_{1}$ .

?

Problem 3.15

Using the description of the dual space $(C(\left[a, b\right]))^{*}$ , prove the linearity and continuity of the given functionals on $C(\left[a, b\right])$ and calculate their norms:

?

(1)

$X = C(\left[a, b\right]), f(x) = \frac{1}{2}(x(a)+x(b))$ ;

(2)

$X = C(\left[0,1\right]), f(x) = x(0)$ ;

(3)

$X = C(\left[-1,1\right]), f(x) = x(0)-2 i x(-1)+(4+3 i) x(1)$ ;

(4)

$X = C(\left[0,1\right]), f(x) = x(0)+\int_{0}^{1} t^{2} x(t) d t$ ;

(5)

$X = C(\left[-1,1\right]), f(x) = \int_{-1}^{0} x(t) d t-\int_{0}^{1} x(t) d t ;$

(6)

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

(7)

$X = C(\left[0,2\right]), f(x) = \int_{0}^{1}(1-2 t) x(t) d t-i x(0)+2 x(2) ;$

(8)

$X = C(\left[0, \pi \right]), f(x) = \int_{0}^{\pi } x(t) \cos t d t ;$

(9)

$X = C(\left[0, \pi \right]), f(x) = \int_{0}^{\pi } x(t)\left|\cos t\right| d t ;$

(10)

$X = C(\left[0, \pi \right]), f(x) = \int_{0}^{\pi } x(t) \cos t d t+i \int_{0}^{\pi } x(t) \sin t d t ;$

(11)

$X = C(\left[0,1\right]), f(x) = \int_{0}^{\ln 2} e^{t} x(t) d t+2 i x(1) ;$

(12)

$X = C(\left[-1,2\right]), f(x) = \int_{-1}^{1}\left|t\right| x(t) d t+3 \int_{1}^{2} x(t) d t-2 x(0) ;$

(13)

$X = C(\left[0,1\right]), f(x) = \sum_{n = 1}^{\infty } \frac{(-1)^{n}}{n^{2}} x\left(\frac{1}{n}\right)$ ;

(14)

$X = C(\left[0,1\right]), f(x) = \sum_{n = 1}^{\infty } \frac{1}{(4 n+1)(4 n+5)} x\left(\frac{1}{n^{2}}\right)$ ;

(15)

$X = C(\left[0,1\right]), f(x) = \sum_{n = 1}^{\infty } \frac{(-1)^{n}}{2^{n}} x\left(1-\frac{1}{n}\right)$ ;

(16)

$X = C(\left[a, b\right]), f(x) = \sum_{k = 1}^{n} \lambda_{k} x\left(t_{k}\right)$ , where $t_{1}, \ldots , t_{n}$ is a set of different points from $\left[a, b\right], \lambda_{1}, \ldots , \lambda_{n}$ is a set of complex numbers;

(17)

$X = C(\left[0,1\right]), f(x) = \int_{0}^{1} x\left(t^{2}\right) d t$ .

Problem 3.16

Let $1 \leq p \leq +\infty$ be fixed. Using the descriptions of the respective dual spaces, prove the linearity and continuity of the given functionals and calculate their norms:

?

(1)

$X = l_{1}, f(x) = \sum_{n = 1}^{\infty } \frac{x_{n}}{n}$ ;

(2)

$X = l_{1}, f(x) = \sum_{n = 1}^{\infty } x_{n} \operatorname {arctg} n$ ;

(3)

$X = l_{2}, f(x) = \sum_{n = 1}^{\infty } \frac{x_{n}}{\sqrt{n(n+1)}}$ ;

(4)

$X = l_{\infty }, f(x) = \sum_{n = 2}^{\infty } \frac{x_{n}}{n^{2}-1}$ ;

(5)

$X = l_{\frac{4}{3}}, f(x) = x_{1}+\sqrt[4]{3} i x_{3}$ ;

(6)

$X = l_{\frac{7}{5}}, f(x) = \sum_{n = 1}^{\infty } \frac{x_{3 n}}{\sqrt[7]{n^{4}}}$ ;

(7)

$X = l_{p}, f(x) = x_{j}$ , where $j \in \mathbb {N}$ is fixed;

(8)

$X = l_{p}, f(x) = 2 i x_{1}+4 x_{2}$ ;

(9)

$X = l_{p}, f(x) = \sum_{n = 1}^{\infty } \frac{x_{n}}{2^{n}}$ ;

(10)

$X = l_{2}, f(x) = \sum_{n = 1}^{\infty } \frac{\sqrt{n}}{2^{n / 2}} x_{n}$ ;

(11)

$X = L_{1}(\mathbb {R}), f(x) = \int_{\mathbb {R}} x(t) \operatorname {arcctg} t d t ;$

(12)

$X = L_{2}(\left[0,2 \pi \right]), f(x) = \int_{0}^{\pi } x(t) e^{i t} \sin t d t ;$

(13)

$X = L_{p}(\left[1,4\right]), f(x) = \int_{1}^{2} t x(t) d t-2 \int_{3}^{4} x(t) d t$ ;

(14)

$X = L_{p}(\mathbb {R}), f(x) = \int_{\mathbb {R}} e^{-t^{2}} x(t) d t ;$

(15)

$X = L_{\infty }([0,+\infty )), f(x) = \int_{0}^{+\infty } \frac{\sin^{2} t}{t^{2}} x(t) d t$ .

Problem 3.17

Let $1 \leq p < +\infty$ be fixed. For which $\alpha \in \mathbb {R}$ the following functionals belong to the respective dual spaces? Calculate the norms of these functionals.

?

(1)

$X = l_{p}, f(x) = \sum_{n = 1}^{\infty } \frac{x_{n}}{n^{\alpha }}$ ;

(2)

$X = l_{p}, f(x) = \sum_{n = 1}^{\infty } x_{n}\left((n+1)^{\alpha }-n^{\alpha }\right)$ ;

(3)

$X = L_{p}(\left[0,1\right]), f(x) = \int_{0}^{1} \frac{x(t)}{t^{\alpha }} d t$ ;

(4)

$X = L_{p}([0,+\infty )), f(x) = \int_{0}^{+\infty } t^{\alpha } e^{-t} x(t) d t$ ;

(5)

$X = L_{p}\left(\left[0, \frac{\pi }{2}\right]\right), f(x) = \int_{0}^{\frac{\pi }{2}} x(t) \sin^{\alpha } t d t$ .

Problem 3.18

(1)(2)

?

(1)

Establish the general form of a linear functional on a finite-dimensional NLS and prove that every linear functional on this space is continuous.

(2)

Describe all subsets $M \subset \mathbb {R}^{2}$ , for which there exists a nonzero linear functional $f$ on $\mathbb {R}^{2}$ such that $\operatorname {Ker} f = M$ . Also solve this problem for subsets of $\mathbb {R}^{3}$ .

Problem 3.19

Prove that $c_{0}^{*} = l_{1}$ , and each continuous linear functional on $c_{0}$ is given by the formula $f(x) = \sum_{n = 1}^{\infty } x_{n} a_{n}, x \in c_{0}$ , where $a = \left(a_{1}, \ldots , a_{n}, \ldots \right) \in l_{1}$ is a fixed element, $\left\| f\right\| = \left\| a\right\|_{1}$ .

?

Problem 3.20

Prove that $c^{*} = l_{1}$ , and each continuous linear functional on $c$ is given by the formula $f(x) = a_{0} \lim_{n \rightarrow \infty } x_{n}+\sum_{n = 1}^{\infty } a_{n} x_{n}, x \in c$ , where $a = \left(a_{0}, a_{1}, \ldots , a_{n}, \ldots \right) \in l_{1}$ is a fixed element, $\left\| f\right\| = \left\| a\right\|_{1}$ .

?

Problem 3.21

Describe all continuous linear functionals on $c_{0}$ , which achieve their norm on $\bar{B}(0,1)$ .

?

Problem 3.22

(1)(2)

?

(1)

Find the general form of a continuous linear functional in a real Banach space $C^{1}(\left[a, b\right])$ .

(2)

Find the general form of the continuous linear functional in a real Banach space $C^{n}(\left[a, b\right])$ , where $n \in \mathbb {N}$ is fixed.

Problem 3.23

(1)(2)

Prove that $f(x) = \int_{a}^{b} p(t) x(t) d t, x \in C(\left[a, b\right])$ is a continuous linear functional, and calculate its norm, if

?

(1)

$p \in C(\left[a, b\right])$ is a fixed function;

(2)

$p \in L_{1}(\left[a, b\right])$ is a fixed function.

Problem 3.24

(1)(2)(3)

Give an example of a functional $f \in C(\left[0,1\right])^{*}$ , for which

?

(1)

there is no element $x_{0} \in C(\left[0,1\right])$ such that $\left\| x_{0}\right\| = 1$ and $f\left(x_{0}\right) = \left\| f\right\|$ ;

(2)

there are infinitely many such elements;

(3)

there is exactly one such element.

Problem 3.25

In the spaces $l_{1}$ and $c$ give examples of continuous linear functionals, which achieve and which do not achieve their norm on the closed unit ball.

?

Problem 3.26

(1)(2)(3)

Let $X$ be an NLS.

?

(1)

Can there exist a functional $f \in X^{*}$ , for which

(a) $\exists !x_{0} \in \bar{B}(0,1):\left|f\left(x_{0}\right)\right| = \left\| f\right\|$ ?

(b) $\exists !x_{0} \in \bar{B}(0,1): f\left(x_{0}\right) = \left\| f\right\|$ ?

(2)

Let $f \in X^{*} \backslash \left\{ 0\right\}$ . Is it correct that

(b) if $x \in \bar{B}(0,1),\left|f(x)\right| = \left\| f\right\|$ , then $\left\| x\right\| = 1$ ?

(3)

Let $f \in X^{*}$ . Consider the following statements:

(b) $\exists x \in \bar{B}(0,1):\left|f(x)\right| = \left\| f\right\|$ ;

(c) $\exists x \in \bar{B}(0,1): f(x) = \left\| f\right\|$ .

Which of these statements are equivalent?

Problem 3.27

Let $X_{1}, X_{2}$ be two normed linear spaces such that $X_{1} \subset X_{2}$ and convergence $x_{n} \rightarrow x$ in $X_{1}$ implies convergence $x_{n} \rightarrow x$ in $X_{2}$ . For a given $f \in X_{2}^{*}$ set $g(x) = f(x), x \in X_{1}$ . Prove that $g \in X_{1}^{*}$ .

?

Problem 3.28

Let $X$ be an NLS, $f \in X^{*} \backslash \left\{ 0\right\} , L: = \left\{ x \in X \mid f(x) = 1\right\}$ . Prove that $\frac{1}{\left\| f\right\| } = \inf_{x \in L}\left\| x\right\|$ .

?

Problem 3.29

Let $X$ be an NLS over the field $\mathbb {K}, f$ be a linear functional on $X$ . Prove that each of the following conditions is equivalent to the continuity of $f$ :

?

(1)

there exists $R > 0$ such that the set $f(\bar{B}(0, R))$ is bounded;

(2)

for any sequence $\left\{ x_{n}: n \geq 1\right\} \subset X$ which converges to zero, the set $\left\{ f\left(x_{n}\right) : n \geq 1\right\}$ is bounded;

(3)

for any Cauchy sequence $\left\{ x_{n}: n \geq 1\right\} \subset X$ , the sequence of numbers $\left\{ f\left(x_{n}\right) : n \geq 1\right\}$ converges in $\mathbb {K}$ .

(4)

for any real $c \in \mathbb {R}$ , the set $\left\{ x \mid f(x) < c\right\}$ is an open set (in the case $\mathbb {K} = \mathbb {R}$ );

(5)

there exists $C > 0$ such that the set $\left\{ x \mid \left|f(x)\right| \leq C\right\}$ has interior points.

Problem 3.30

Let $X$ be an NLS over the field $\mathbb {K}$ , and $f$ be a linear unbounded functional on $X$ . Prove that $f$ attains all values from the field $\mathbb {K}$ in any neighborhood of zero. Is it true that $f$ attains all values from the field $\mathbb {K}$ in any neighborhood of any point?

?

Problem 3.31

Let $X$ be an NLS over the field $\mathbb {K}$ , and $f, g \in X^{*}$ be such that $\operatorname {Ker} f \subset \operatorname {Ker} g$ . Prove that there exists $\alpha \in \mathbb {K}$ such that $g = \alpha f$ .

?

Problem 3.32

Let $X$ be an NLS, $f, f_{1}, \ldots , f_{n}$ be linear functionals on $X$ . Prove that $f$ is a linear combination of functionals $f_{1}, \ldots , f_{n}$ if and only if

\bigcap _{k = 1}^{n} \operatorname {Ker} f_{k} \subset \operatorname {Ker} f.

?

Problem 3.33

(1)(2)

In which normed linear spaces $X$ does there exist a functional $f \in X^{*}$

?

(1)

(2)

such that $\operatorname {Ker} f = \left\{ 0\right\}$ ?

Problem 3.34

Let there be functionals $f, g \in X^{*}$ on an NLS $X$ such that $\operatorname {Ker} f \cap \operatorname {Ker} g = \left\{ 0\right\}$ . Prove that $\operatorname {dim} X \leq 2$ .

?

Problem 3.35

Let $X$ be an NLS over the field $\mathbb {K}, f$ be a nonzero linear functional on $X$ . Prove that $f$ is an open mapping, i.e. the image of any open set in $X$ under the mapping $f$ is an open set in $\mathbb {K}$ .

?

Problem 3.36

Let $X$ be an NLS, $f$ be a linear functional on $X$ . Prove that $f \in X^{*}$ if and only if $\operatorname {Ker} f$ is a closed set.

?

Problem 3.37

(1)(2)

Let $X$ be an NLS over the field $\mathbb {K}$ , $f$ be a linear functional on $X$ . Prove that

?

(1)

$\operatorname {Ker} f$ is closed in $X$ or is a dense set in $X$ ;

(2)

if the set $\left\{ x \in X \mid f(x) = c_{0}\right\}$ is closed for some $c_{0} \in \mathbb {K}$ , then the set $\left\{ x \in X \mid f(x) = c\right\}$ is closed for every $c \in \mathbb {K}$ .

Problem 3.38

Let $X$ be an NLS, $f \in X^{*}$ . Prove that $\left|f(x)\right| = \left\| f\right\| \cdot \rho (x, \operatorname {Ker} f)$ for all $x \in X$ , where $\rho (x, \operatorname {Ker} f) = \inf_{y \in \operatorname {Ker} f}\left\| x-y\right\|$ .

?

Problem 3.39

Let $X$ be an NLS over the field $\mathbb {K} = \mathbb {R}, f \in X^{*}$ , and elements $x_{0}, y_{0} \in X$ be such that $f\left(x_{0}\right) \cdot f\left(y_{0}\right) < 0$ . Prove that

\left\| x_{0}-y_{0}\right\| \geq \rho \left(x_{0}, \operatorname {Ker} f\right)+\rho \left(y_{0}, \operatorname {Ker} f\right)

?

Problem 3.40

(1)(2)

Let $X$ be an NLS, $f \in X^{*}, f \neq 0$ .

?

(1)

Prove that the following statements are equivalent:

(a) there exists an element $z \in \bar{B}(0,1)$ such that $\left|f(z)\right| = \left\| f\right\|$ ;

(b) for every $x \in X$ there exists $y \in \operatorname {Ker} f$ such that $\rho (x, \operatorname {Ker} f) = \left\| x-y\right\|$ (i.e. $y$ is the element of the best approximation for $x$ in $\operatorname {Ker} f$ );

(c) for some $x \in X \backslash \operatorname {Ker} f$ there exists $y \in \operatorname {Ker} f$ such that $\rho (x, \operatorname {Ker} f) = \left\| x-y\right\|$ .

(2)

Under what conditions on the functional $f$ for every element $x \in X$ there exists no more than one element of the best approximation in $\operatorname {Ker} f$ ?

Problem 3.41

(1)(2)

Give examples of subspaces $L$ in $X = C(\left[0,1\right])$ and in $X = c_{0}$ such that

?

(1)

for any $x_{0} \notin L$ there is no element of the best approximation in $L$ ;

(2)

for any $x_{0} \notin L$ there exist infinitely many elements of the best approximation in $L$ .

Problem 3.42

(1)(2)(3)

A linear functional $f$ on the space $C(\left[a, b\right])$ is called nonnegative, if for every $x \in C(\left[a, b\right])$ such that $x(t) \geq 0, t \in [a, b]$ , the inequality $f(x) \geq 0$ holds. Prove the following statements.

?

(1)

A nonnegative functional $f$ on a real space $C(\left[a, b\right])$ is continuous, and $\left\| f\right\| = f\left(e_{0}\right)$ , where $e_{0}(t) = 1, t \in [a, b]$ .

(2)

The statement of item (1) is valid in a complex space $C(\left[a, b\right])$ as well.

(3)

If a functional $f \in (C(\left[a, b\right]))^{*}$ is such that $\left\| f\right\| = f\left(e_{0}\right)$ , then $f$ is nonnegative.

Problem 3.43

Establish the necessary and sufficient condition on the function $g \in B V_{0}(\left[a, b\right])$ , under which the functional $f(x) = \int_{a}^{b} x(t) d g(t)$ is nonnegative.

?

Problem 3.44

Prove that for every functional $f \in (C(\left[a, b\right]))^{*}$ , where $C(\left[a, b\right])$ is a real space, there exist nonnegative functionals $f_{1}$ and $f_{2}$ such that $f = f_{1}-f_{2}$ .

?

Problem 3.45

(1)(2)(3)

A linear functional $f$ on a real space $C(\left[a, b\right])$ is called multiplicative if $f(x y) = f(x) f(y), x, y \in C(\left[a, b\right])$ . Prove that

?

(1)

for a fixed $\tau \in [a, b]$ the functional $f(x) = x(\tau ), x \in C(\left[a, b\right])$ , is multiplicative;

(2)

every nonzero multiplicative functional is continuous and has a norm equal to one;

(3)

if $f$ is a nonzero multiplicative functional, then there exists $\tau \in [a, b]$ such that $f(x) = x(\tau ), x \in C(\left[a, b\right])$ .

Problem 3.46

Let $X_{1}, X_{2}$ be two normed linear spaces, $X: = X_{1} \bigoplus X_{2}$ be their direct sum with the norm

\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 .

Prove that every functional $h \in X^{*}$ is uniquely represented in the form

h((x, y)) = f_{1}(x)+f_{2}(y), \quad (x, y) \in X,

where $f_{i} \in X_{i}^{*}, i = 1,2$ .

?

Problem 3.47

Prove that in an infinite-dimensional NLS there exist linear discontinuous functionals.

?

Problem 3.48

Let $X$ be an NLS, $M \subset X$ be a convex dense subset in $X$ and $f \in X^{*}$ . Prove that the set $M \cap \operatorname {Ker} f$ is dense in $\operatorname {Ker} f$ .

?

Problem 3.49

(Yamabe) Let $X$ be an NLS, $M \subset X$ be a convex dense subset in $X, x \in X, \varepsilon > 0, f_{1}, \ldots , f_{n} \in X^{*}$ . Prove that $\exists y \in M,\left\| x-y\right\| < \varepsilon : f_{i}(y) = f_{i}(x), 1 \leq i \leq n$ .

?

Problem 3.50

Prove that for any function $x \in C(\left[a, b\right])$ , any points $t_{1}, \ldots , t_{n} \in [a, b]$ and any $\varepsilon > 0$ there exists a polynomial $p$ such that $\max_{t \in [a, b]}\left|x(t)-p(t)\right| < \varepsilon$ , and $p\left(t_{i}\right) = x\left(t_{i}\right), 1 \leq i \leq n$ .

?