Section 20: Problem 10 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $X$ denote the subset of $\mathbb{R}^{\omega}$ consisting of all sequences $(x_{1},x_{2},\ldots)$ such that $\sum x_{i}^{2}$ converges. (You may assume the standard facts about infinite series. In case they are not familiar to you, we shall give them in Exercise 11 of the next section.)

(a) Show that if $\mathbf{x},\mathbf{y}\in X$ , then $\sum|x_{i}y_{i}|$ converges. [Hint: Use (b) of Exercise 9 to show that the partial sums are bounded.]

(b) Let $c\in\mathbb{R}$ . Show that if $\mathbf{x},\mathbf{y}\in X$ , then so are $\mathbf{x}+\mathbf{y}$ and $c\mathbf{x}$ .

(c) Show that $d(\mathbf{x},\mathbf{y})=\left[\sum_{i=1}^{\infty}(x_{i}-y_{i})^{2}\right]^{1/2}$ is a well-defined metric on $X$ .

(a) Using the hint and Exercise 9(b), for any $\mathbf{z}$ , let $\mathbf{z}'=(|z_{1}|,|z_{2}|,\ldots)$ and $\mathbf{z}_{n}=(z_{1},\ldots,z_{n})$ , then $\Vert\mathbf{z}'\Vert=\Vert\mathbf{z}\Vert$ , and $\sum_{i=1}^{n}|x_{i}y_{i}|=$ $|\mathbf{x}'_{n}\cdot\mathbf{y}'_{n}|\le$ $\Vert\mathbf{x}'_{n}\Vert\Vert\mathbf{y}'_{n}\Vert=$ $\Vert\mathbf{x}_{n}\Vert\Vert\mathbf{y}_{n}\Vert$ , and the expression on the right converges.

(b) $\sum(x_{i}+y_{i})^{2}=$ $\sum|x_{i}+y_{i}|^{2}\le$ $\sum(|x_{i}|+|y_{i}|)^{2}=$ $\sum x_{i}^{2}+\sum y_{i}^{2}+2\sum|x_{i}y_{i}|$ which converges by (a). Alternatively, it follows immediately from Exercise 9(c) (the partial sums are bounded). $\sum c^{2}x_{i}^{2}=c^{2}\sum x_{i}^{2}$ also converges.

(c) It is. The sum converges by (b), it is nonnegative, and equals zero iff $x_{i}=y_{i}$ for every $i\in\mathbb{Z}_{+}$ . The symmetry is clear as well. Further, the triangle inequality holds by Exercise 9(c) (used for partial sums).

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Section 20: Problem 10 Solution

Section 20: Problem 10 Solution