3.6.E: Problems on Normed Linear Spaces (Exercises)
Show that distances in normed spaces obey the laws stated in Theorem 5 of §§1-3.
Complete the proof of assertions made in Example (C) and Note 2.
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 \((2) ?\)
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
\[
\bigcap_{n=1}^{\infty} L\left[a_{n}, b_{n}\right] \neq \emptyset.
\]
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. \(]\)
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. \(]\)
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. \(]\)
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 \(E\) ).