Chapter 5.1, Problem 18E

Here is an infinite-dimensional version of Euclidean space: In the space of all infinite sequences, consider the subspace ${\ell }_2$ , of square-summable sequences [i.e., those sequences $\left(x_1,x_2,\mathrm{...}\right)$ for which the infinite series $x_1^2+x_2^2+\cdots$ converges]. For $\stackrel{\to }{x}$ and $\stackrel{\to }{y}$ in ${\ell }_2$ , we define
   $\Vert \stackrel{\to }{x}\Vert =\sqrt{x_1^2+x_2^2+\cdots },\stackrel{\to }{x}\cdot \stackrel{\to }{y}=x_1{y}_1+x_2{y}_2+\cdots .$
(Why does the series $x_1{y}_1+x_2{y}_2+\cdots$ converge?)
a. Check that $\stackrel{\to }{x}=\left(1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\cdots \right)$ is in ${\ell }_2$ , and find $\Vert \stackrel{\to }{x}\Vert$ . Recall the formula for the geometric series: $1+a+a^2+a^3+\cdots =1/\left(1-a\right)$ , if $-1<a<1$ .
b. Find the angle between $\left(1,0,0,\mathrm{...}\right)$ and $\left(1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\mathrm{...}\right)$ .
c. Give an example of a sequence $\left(x_1,x_2,\mathrm{...}\right)$ that converges to 0 (i.e. $\underset{n\to 0}{\lim }{x}_n=0$ ) hut does not belong to ${\ell }_2$ .
d. Let L be the subspace of ${\ell }_2$ spanned by $\left(1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\mathrm{...}\right)$ . Find the orthogonal projection of $\left(1,0,0,\mathrm{...}\right)$ onto L.
The Hubert space ${\ell }_2$ was initially used mostly in physics: Werner Heisenberg’s formulation of quantum mechanics is in terms of ${\ell }_2$ . Today, this space is used in many other applications, including economics. See, for example, the work of the economist Andreu Mas-Colell of the University of Barcelona.