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# 3.6.E: Problems on Normed Linear Spaces (Exercises)

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Exercise $$\PageIndex{1}$$

Show that distances in normed spaces obey the laws stated in Theorem 5 of §§1-3.

Exercise $$\PageIndex{2}$$

Complete the proof of assertions made in Example (C) and Note 2.

Exercise $$\PageIndex{3}$$

Define $$|x|=x_{1}$$ for $$x=\left(x_{1}, \ldots, x_{n}\right)$$ in $$C^{n}$$ or $$E^{n} .$$ Is this a norm? Which (if any) of the laws (i) - (iii) does it obey? How about formula $$(2) ?$$

Exercise $$\PageIndex{4}$$

Do Problem 3 in §§4-6 for a general normed space $$E,$$ with lines defined as in $$E^{n}$$ (see also Problem 7 in §9). Also, show that contracting sequences of line segments in $$E$$ are $$f$$-images of contracting sequences of intervals in $$E^{1} .$$ Using this fact, deduce from Problem 11 in Chapter 2 §§8-9, an analogue for line segments in $$E$$, namely, if
$L\left[a_{n}, b_{n}\right] \supseteq L\left[a_{n+1}, b_{n+1}\right], \quad n=1,2, \dots$
then
$\bigcap_{n=1}^{\infty} L\left[a_{n}, b_{n}\right] \neq \emptyset.$

Exercise $$\PageIndex{5}$$

Take for granted the lemma that
$a^{1 / p} b^{1 / q} \leq \frac{a}{p}+\frac{b}{q}$
if $$a, b, p, q \in E^{1}$$ with $$a, b \geq 0$$ and $$p, q>0,$$ and
$\frac{1}{p}+\frac{1}{q}=1.$
(A proof will be suggested in Chapter 5, §6, Problem $$11 . )$$ Use it to prove Hölder's inequality, namely, if $$p>1$$ and $$\frac{1}{p}+\frac{1}{q}=1,$$ then
$\sum_{k=1}^{n}\left|x_{k} y_{k}\right| \leq\left(\sum_{k=1}^{n}\left|x_{k}\right|^{p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n}\left|y_{k}\right|^{q}\right)^{\frac{1}{q}} \text { for any } x_{k}, y_{k} \in C.$
[Hint: Let
$A=\left(\sum_{k=1}^{n}\left|x_{k}\right|^{p}\right)^{\frac{1}{p}} \text { and } B=\left(\sum_{k=1}^{n}\left|y_{k}\right|^{q}\right)^{\frac{1}{q}}.$
If $$A=0$$ or $$B=0,$$ then all $$x_{k}$$ or all $$y_{k}$$ vanish, and the required inequality is trivial. Thus assume $$A \neq 0$$ and $$B \neq 0 .$$ Then, setting
$a=\frac{\left|x_{k}\right|^{p}}{A^{p}} \text { and } b=\frac{\left|y_{k}\right|^{q}}{B^{q}}$
in the lemma, obtain
$\frac{\left|x_{k} y_{k}\right|}{A B} \leq \frac{\left|x_{k}\right|^{p}}{p A^{p}}+\frac{\left|y_{k}\right|^{q}}{q B^{q}}, k=1,2, \ldots, n.$
Now add up these $$n$$ inequalities, substitute the values of $$A$$ and $$B,$$ and simplify. $$]$$

Exercise $$\PageIndex{6}$$

Prove the Minkowski inequality,
$\left(\sum_{k=1}^{n}\left|x_{k}+y_{k}\right|^{p}\right)^{\frac{1}{p}} \leq\left(\sum_{k=1}^{n}\left|x_{k}\right|^{p}\right)^{\frac{1}{p}}+\left(\sum_{k=1}^{n}\left|y_{k}\right|^{p}\right)^{\frac{1}{p}}$
for any real $$p \geq 1$$ and $$x_{k}, y_{k} \in C$$.
[Hint: If $$p=1,$$ this follows by the triangle inequality in $$C .$$ If $$p>1$$ , let
$A=\sum_{k=1}^{n}\left|x_{k}+y_{k}\right|^{p} \neq 0.$
(If $$A=0$$, all is trivial.) Then verify (writing "$$\sum$$" for "$$\sum_{k=1}^{n}$$" for simplicity)
$A=\sum\left|x_{k}+y_{k}\right|\left|x_{k}+y_{k}\right|^{p-1} \leq \sum\left|x_{k}\right|\left|x_{k}+y_{k}\right|^{p-1}+\sum\left|y_{k}\right|\left|x_{k}+y_{k}\right|^{p-1}$
Now apply Hölder's inequality (Problem 5) to each of the last two sums, with $$q=$$ $$p /(p-1),$$ so that $$(p-1) q=p$$ and $$1 / p=1-1 / q .$$ Thus obtain
$A \leq\left(\sum\left|x_{k}\right|^{p}\right)^{\frac{1}{p}}\left(\sum\left|x_{k}+y_{k}\right|^{p}\right)^{\frac{1}{q}}+\left(\sum\left|y_{k}\right|^{p}\right)^{\frac{1}{p}}\left(\sum\left|x_{k}+y_{k}\right|^{p}\right)^{\frac{1}{q}}.$
Then divide by $$A^{\frac{1}{q}}=\left(\sum\left|x_{k}+y_{k}\right|^{p}\right)^{\frac{1}{q}}$$ and simplify. $$]$$

Exercise $$\PageIndex{7}$$

Show that Example (B) indeed yields a norm for $$C^{n}$$ and $$E^{n}$$.
[Hint: For the triangle inequality, use Problem $$6 .$$ The rest is easy. $$]$$

Exercise $$\PageIndex{8}$$

A sequence $$\left\{x_{m}\right\}$$ of vectors in a normed space $$E\left(\text { e.g. }, \text { in } E^{n} \text { or } C^{n}\right)$$ is said to be bounded iff
$\left(\exists c \in E^{1}\right)(\forall m) \quad\left|x_{m}\right|<c,$
i.e., iff $$\sup _{m}\left|x_{m}\right|$$ is finite.
Denote such sequences by single letters, $$x=\left\{x_{m}\right\}, y=\left\{y_{m}\right\},$$ etc. and define
$x+y=\left\{x_{m}+y_{m}\right\}, \text { and } a x=\left\{a x_{m}\right\} \text { for any scalar } a.$
Also let
$|x|=\sup _{m}\left|x_{m}\right|.$
Show that, with these definitions, the set $$M$$ of all bounded infinite sequences in $$E$$ becomes a normed space (in which every such sequence is to be treated as a single vector, and the scalar field is the same as that of $$E$$ ).