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Chapter 3: Problem 22

If \(0-1}, \quad x_{n}(m)=0 \quad \text { if } m \neq n $$ Let \(K\) consist of \(0, x_{1}, x_{2}, x_{3} \ldots .\) If $$ y_{N}-N^{-1}\left(x_{1}+\cdots+x_{s}\right) $$ show that \(\left\\{y_{N}\right\\}\) is unbounded in \(\ell ?\)

Short Answer

Expert verified
To summarize the solution, we defined a sequence of points \(x_n\), created a compact set \(K\) containing these points along with the origin, and then defined a sequence \(y_N\) as an averaging of the first \(s\) elements of the sequence \(x_n\). Using the p-norm, we proved that the sequence \(y_N\) is unbounded. Finally, since the sequence \(y_N\) can be expressed as a linear combination of the points in the compact set \(K\), we concluded that the convex hull of \(K\) is unbounded.

Step by step solution

01

Define the sequence x_n and compact set K

To begin with, let's define the sequence of points \(x_n\) as described in the exercise. For each positive integer \(n\), let \(x_n \in \ell\) be defined as follows: $$ x_n(m)= \begin{cases} n^{-1}, & \text{if} \ m=n \\ 0, & \text{otherwise} \end{cases} $$ Next, we define the compact set \(K\) to be a collection of all \(x_n\) including the origin of the sequence. Hence, \(K=\{0, x_1, x_2, x_3, ...\}\).
02

Define the sequence y_N

Now let's define a new sequence \(y_N\) as follows: $$ y_N=N^{-1}(x_1+x_2+...+x_s) $$ where \(s\) is any positive integer. Our aim is to prove that the sequence \(\{y_N\}\) is unbounded in \(\ell\).
03

Relation between y_N and x_n

Notice that the relation between \(y_N\) and \(x_n\) can be expressed as: $$ y_N(m)= \begin{cases} N^{-1}s^{-1}, & \text{if} \ m \leq s \\ 0, & \text{otherwise} \end{cases} $$ So, the sequence \(\{y_N\}\) is an averaging of the first \(s\) elements of the sequence \(\{x_n\}\) with a factor of \(N^{-1}\).
04

Prove that y_N is unbounded

Let's denote the p-norm of a vector \(x \in \ell\) as \(\|x\|_p\). Also note that the p-norm is homogeneous, meaning that \(\|\alpha x\|_p = |\alpha| \|x\|_p\) for any scalar \(\alpha\) and vector \(x\). In order to show that the sequence \(\{y_N\}\) is unbounded, we will prove that \(\|y_N\|_p\) can be arbitrarily large for some appropriate choices of \(N\) and \(s\). Using the definition of the p-norm, we have: $$ \|y_N\|_p^p = \sum_{m=1}^s \left(N^{-1}s^{-1}\right)^p = s\left(N^{-1}s^{-1}\right)^p $$ Now, let \(\epsilon > 0\) be any real number. Setting \(N=s^{1/p}(\epsilon^{1/p}-1)\), we find: $$ \|y_N\|_p^p = s\left(\frac{s^{1/p}(\epsilon^{1/p}-1)}{s}\right)^p = s\left(\epsilon^{1/p}-1\right)^p > s(\epsilon-1) $$ As \(s\) can be chosen arbitrarily large, we can make \(\|y_N\|_p^p\) as large as we desire, implying that \(\{y_N\}\) is indeed unbounded.
05

Convex Hull of K is unbounded

The sequence \(\{y_N\}\) can be expressed as a linear combination of the sequence \(\{x_n\}\), since \(y_N=N^{-1}(x_1+x_2+...+x_s)\). Therefore, \(\{y_N\}\) is included in the convex hull of the compact set \(K\). Since we proved that the sequence \(\{y_N\}\) is unbounded, it follows that the convex hull of the compact set \(K\) is also unbounded. This concludes the proof.

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Most popular questions from this chapter

Suppose \(L^{2}=L^{2}([-1,1])\), with respect to Lebesgue measure. For each scalar \(\alpha\), let \(E_{2}\) be the set of all continuous functions \(f\) on \([-1,1]\) such that \(f(0)=\alpha\). Show that cach \(E_{x}\) is convex and that each is dense in \(L^{2}\). Thus \(E_{x}\) and \(E_{\mu}\) are disjoint convex sets (if \(\alpha \neq \beta\) ) which cannot be separated by any continuous linear functional \(A\) on \(L^{2} .\) Hint: What is \(A\left(E_{2}\right)\) ?

Let \(\ell^{\infty}\) be the space of all real bounded functions \(x\) on the positive integers. Let \(\tau\) be the translation operator defined on \(f^{*}\) by the equation $$ (\tau x)(n)=x(n+1) \quad(n=1,2,3, \ldots) $$ Prove that there exists a linear functional \(\Lambda\) on \(\ell^{-\infty}\) (called a Banach limit) such that (a) \(\Lambda \tau x-A x\), and (b) \(\underset{n \rightarrow \infty}{\lim \inf } x(n) \leq \Lambda x \leq \lim _{n \rightarrow \infty} \sup x(n)\) for every \(x \in \ell^{\circ}\). Suggestion: Define $$ \begin{aligned} &\begin{aligned} \mathrm{A}_{\alpha} x &=\frac{x(1)+\cdots+x(n)}{n} \\ M &=\left\\{x \in \ell^{\propto}: \lim _{n \rightarrow \infty} \Lambda_{n} x=\Lambda x \text { exists }\right\\} \\ p(x) &=\lim \sup A_{n} x \end{aligned} \end{aligned} $$ and apply Theorem \(3.2 .\)

Suppose \(K\) is a compact convex set in \(R^{n}\). Prove that every \(x \in K\) is a convex combination of at most \(n+1\) extreme points of \(K\). Suggestion: Use induction on \(n\). Draw a line from some extreme point of \(K\) through \(x\) to where it leaves \(K\). Use Exercise \(1 .\)

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Suppose a topological vector space \(X\) contains a countable set \(E=\left\\{e_{1}, e_{2}, e_{3}, \ldots\right\\}\) with the following properties: (a) \(c_{n} \rightarrow 0\) as \(n \rightarrow \infty\). (b) Every \(x \in X\) is a finite linear combination of members of \(E, x=\sum \gamma_{n}(x) e_{z}\) (c) No \(e_{n}\) is in the closed subspace of \(X\) generated by the other \(e_{t}\). For example, \(X\) could be the space of all complex polynomials $$ f(z)=a_{0}+a_{1} z+\cdots+a_{n} z^{n} $$ with norm $$ \|f\|-\left\\{\int_{-=}^{*}\left|f\left(e^{i \theta}\right)\right|^{2} d \theta\right\\}^{1 / 2} $$ and with \(e_{n}(z)=n^{-1} z^{n-1}(n=1,2,3, \ldots)\). Prove that each \(\gamma_{n}\) in \((b)\) is in \(X^{*}\). Put \(K=E \cup\\{0\\}\). Then \(K\) is compact. Prove that the convex hull \(H\) of \(K\) is closed but not compact and that the extreme points of \(H\) are exactly the points of \(K\).
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