Math Problems Vault

Problem 2.1

(a)(b)(c)

?

(a)

Prove Young's inequality: $a b \leqslant \frac{a^{p}}{p}+\frac{b^{q}}{q}$ , where $a \geqslant 0, b \geqslant 0$ , $p > 1, q > 1$ and $\frac{1}{p}+\frac{1}{q} = 1$ . Prove that equality is attained if and only if $b = a^{p-1}$ .

(b)

Let $(M, \Sigma , \mu )$ be a measure space with a sigma-finite measure, and let the measurable functions $x$ and $y$ be integrable on $M$ with respect to the measure $\mu$ to the powers $1 < p < \infty$ and $1 < q < \infty$ respectively, $\frac{1}{p}+\frac{1}{q} = 1$ . Prove the integral Hölder inequality

\int _{M}|x(t) y(t)| d \mu \leqslant \left(\int _{M}|x(t)|^{p} d \mu \right)^{1 / p}\left(\int _{M}|y(t)|^{q} d \mu \right)^{1 / q} \tag {2.1}

Prove that equality is attained if and only if the functions $\left|x\right|^{p}$ and $\left|y\right|^{q}$ are linearly dependent.

(c)

Let $(M, \Sigma , \mu )$ be a measure space with a sigma-finite measure, and let the measurable functions $x$ and $y$ be integrable on $M$ with respect to the measure $\mu$ to the power $1 \leqslant p < \infty$ . Prove the integral Minkowski inequality

\left(\int _{M}|x(t)+y(t)|^{p} d t\right)^{1 / p} \leqslant \left(\int _{M}|x(t)|^{p} d t\right)^{1 / p}+\left(\int _{M}|y(t)|^{p} d t\right)^{1 / p} \tag {2.2}

Prove that for $p > 1$ equality in the integral Minkowski inequality is attained if and only if the functions $x$ and $y$ are linearly dependent.

Problem 2.2

(a)(b)

Devise measure spaces $(M, \Sigma , \mu )$ for which inequalities (2.1) and (2.2) turn into the numerical Hölder and Minkowski inequalities $(a_{k}, b_{k} \in \mathbb {C})$ :

?

(a)

\sum _{k = 1}^{n(\infty )}\left|a_{k} b_{k}\right| \leqslant \left(\sum _{k = 1}^{n(\infty )}\left|a_{k}\right|^{p}\right)^{1 / p}\left(\sum _{k = 1}^{n(\infty )}\left|b_{k}\right|^{q}\right)^{1 / q}

(b)

\left(\sum _{k = 1}^{n(\infty )}\left|a_{k}+b_{k}\right|^{p}\right)^{1 / p} \leqslant \left(\sum _{k = 1}^{n(\infty )}\left|a_{k}\right|^{p}\right)^{1 / p}+\left(\sum _{k = 1}^{n(\infty )}\left|b_{k}\right|^{p}\right)^{1 / p}

Problem 2.3

(a)(b)

Give examples of measurable functions $x$ such that:

?

(a)

$x \notin L_{\infty }[0,1]$ ;

(b)

$x$ belongs to $L_{\infty }[0,1]$ , but is not bounded on $[0,1]$ .

Problem 2.4

For functions $x \in L_{1}(M, \Sigma , \mu ), y \in L_{\infty }(M, \Sigma , \mu )$ prove the inequality $\int_{M}\left|x(t) y(t)\right| d \mu \leqslant \left\| x\right\|_{1}\left\| y\right\|_{\infty }$ .

?

Problem 2.5

Prove that if $x \in L_{\infty }(M, \Sigma , \mu )$ and the measure $\mu$ is finite, then $\lim_{p \rightarrow \infty }\left\| x\right\|_{p} = \left\| x\right\|_{\infty }$ . In particular, for any vector $x \in \mathbb {C}^{n}$ it holds that

\lim _{p \rightarrow \infty }\left(\sum _{k = 1}^{n}\left|x_{k}\right|^{p}\right)^{1 / p} = \max _{1 \leqslant k \leqslant n}\left|x_{k}\right|

?

Problem 2.6

Verify the axioms of a norm in the spaces (see the list of spaces):

?

(a)

$L_{\infty }(M, \Sigma , \mu )$ ;

(b)

$C[0,1]$ ;

(c)

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

(d)

$B C(\mathbb {R})$ ;

(e)

$\mathcal{A C}(D)$ ;

(f)

$C_{0}(\mathbb {R})$ ;

(g)

$C_{p}[0,1], p \in [1, \infty )$ ;

(h)

$C_{2}^{1}[0,1]$ .

Problem 2.7

Prove that the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ , introduced on the space $C^{n}[a, b]$ in the list of spaces, satisfy the axioms of a norm and are equivalent.

?

Problem 2.8

Among the mappings of the linear space of continuously differentiable functions on the segment [ 0,1 ] listed below, choose the mappings satisfying the axioms of a norm, and order these norms with respect to the subordination relation:

?

(a)

$\left\| x\right\|_{0} = \max_{t \in [0,1]}\left|x(t)\right|$ ;

(b)

$\left\| x\right\|_{1} = \max_{t \in [0,1]}\left|x^{\prime }(t)\right|$ ;

(c)

$\left\| x\right\|_{2} = \max_{t \in [0,1]}\left|x(t)\right|+\max_{t \in [0,1]}\left|x^{\prime }(t)\right|$ ;

(d)

$\left\| x\right\|_{3} = \max_{t \in [0,1]}\left|x^{\prime }(t)\right|+\left|x(a)\right|$ , where $a \in [0,1]$ ;

(e)

$\left\| x\right\|_{4} = \max_{t \in [0,1]}\left|x^{\prime }(t)\right|+\int_{0}^{1}\left|x(t)\right| d t$ ;

(f)

$\left\| x\right\|_{5} = \max_{t \in [0,1]}\left|x^{\prime }(t)\right|+\left|x(a)-x(b)\right|$ , where $a, b \in [0,1]$ .

Problem 2.9

Let the function $v(t)$ be continuous and positive on $[0,1]$ . Prove that the mapping $F: x \mapsto \left(\int_{0}^{1} v(t)|x(t)|^{p} d t\right)^{1 / p}$ , acting from $L_{p}[0,1]$ to $\mathbb {R}$ , where $p \in [1, \infty )$ , satisfies the axioms of a norm, and this norm is equivalent to the standard $\left\| \cdot \right\|_{p}$ .

?

Problem 2.10

Verify the axioms of a norm on the space of functions of bounded variation $B V[0,1]$ .

?

Problem 2.11

Prove that all norms $\left\| x\right\|_{a} = \operatorname {Var}_{0}^{1} x(t)+\left|x(a)\right|$ , where $0 \leqslant a \leqslant 1$ , on the space of functions of bounded variation $B V[0,1]$ are equivalent to each other and equivalent to the norm $\left\| \cdot \right\|_{B V}$ from the list of spaces.

?

Problem 2.12

Verify the axioms of a norm in the space $B V_{0}[0,1]$ (see the list of spaces).

?

Problem 2.13

Prove Theorem 2.1 on the equivalence of norms in a finite-dimensional space.

Theorem 2.1: If $\dim X < \infty$ (here $\dim X$ is the linear dimension of the linear space $X$ ) and $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ are two different norms on $X$ , then $\left\| \cdot \right\|_{1} \sim \left\| \cdot \right\|_{2}$ , i.e., there exist constants $c, C > 0$ such that $c\left\| x\right\|_{1} \leqslant \left\| x\right\|_{2} \leqslant C\left\| x\right\|_{1}$ for all $x \in X$ .

?

Problem 2.14

Let $p$ be a numerical function on a linear space $X$ satisfying the first two axioms of a norm. Prove that the third axiom (the triangle inequality) is equivalent to the convexity of the set $\left\{ x \in X: p(x) \leqslant 1\right\}$ .

?

Problem 2.15

Prove that the function $\left\| x\right\|_{p} = \left(\sum_{k = 1}^{n}\left|x_{k}\right|^{p}\right)^{1 / p}$ is not a norm for $0 < p < 1$ on an $n$ -dimensional linear space for $n \geqslant 2$ . $p \leqslant q$ .

?

Problem 2.16

Find $\inf_{x \neq 0} \frac{\left\| x\right\|_{p}}{\left\| x\right\|_{q}}$ and $\sup_{x \neq 0} \frac{\left\| x\right\|_{p}}{\left\| x\right\|_{q}}$ , where $x \in \mathbb {R}^{n}$ , for all $p$ and $q \in [1, \infty ]$ ,

?

Problem 2.17

Prove that in the linear space $l_{p}, 1 \leqslant p \leqslant \infty$ , the norms $\left\| \cdot \right\|_{p}$ and $\left\| \cdot \right\|_{q}, q > p$ , are not equivalent, but $\left\| \cdot \right\|_{q} \prec \left\| \cdot \right\|_{p}$ .

?

Problem 2.18

Prove that the norms $\left\| \cdot \right\|_{p}$ and $\left\| \cdot \right\|_{q}, 1 \leqslant p < q \leqslant \infty$ , in the linear space of continuous functions on $[0,1]$ are not equivalent, but $\left\| \cdot \right\|_{p} \prec \left\| \cdot \right\|_{q}$ .

?