## Metric Spaces

July 2020

Section 20, Problem 8

Let $$X$$ be the subset of $$\mathbb{R}^{\omega}$$ consisting of all sequences such that $$\sum x^2_i$$ converges. Then the formula

$d(\textbf{x}, \textbf{y}) = \left[\sum\limits_{i=1}^{\infty} (x_i - y_i)^2\right]^{1/2}$

defines a metric on $$X$$. $$X$$ can inheret a subspace topology from the box, product, and uniform topologies on $$\mathbb{R}^{\omega}$$ and it also has the topology given by $$d$$, called the $$\ell^2$$-topology.

• (a) Show that on $$X$$, we have inclusions $$\textrm{box topology } \supset \ell^2 \textrm{-topology } \supset \textrm{ uniform topology}$$

• (b) The set $$\mathbb{R}^{\infty}$$ of all sequences (actually, $$\omega$$-tuples) that are zero for all but finitely many values is contained in $$X$$. Show that the four topologies $$\mathbb{R}^{\infty}$$ can inherit from $$X$$ are all distinct.

• (c) The set $$H = \prod\limits_{n \in \mathbb{Z}_+}[0, 1/n]$$ is contained in $$X$$ and called the Hilbert cube. Compare the four topologies that $$H$$ inherits as a subspace of $$X$$.

*This is a pretty long problem (meaning it's a bit of a pain to write out the solutions), and so I will return to finish the other two parts at another time.

(a) Given the standard bounded metric on $$\mathbb{R}$$, $$\overline{d}$$, the uniform metric which induces the uniform topology on $$\mathbb{R}^{\omega}$$ is

$\overline{p}(\textbf{x, y}) = \textrm{sup}\{\overline{d}(x_{\alpha}, y_{\alpha}) | \alpha \in \mathbb{Z}_+\}$

We assert that for every point $$\textbf{x}$$ of every basis element $$B_u$$ of the uniform topology, which we shall denote by $$X_u$$, there is a basis element $$B_{\ell}$$ of the $$\ell^2$$-topology, which we shall denote by $$X_{\ell}$$ such that $$\textbf{x} \in B_{\ell} \subset B_u$$.

Let $$B_{\overline{p}}(\textbf{y}, \epsilon)$$ be the $$\epsilon$$-ball centered at some $$\textbf{y} \in X$$ for some $$\epsilon > 0$$. Then, given any $$\textbf{x} \in B_{\overline{p}}(\textbf{y}, \epsilon)$$, we can define the $$\delta$$-ball $$B_d(\textbf{x}, \delta)$$ such that $$\overline{p}(\textbf{x, y}) + \delta < \epsilon$$. Such a ball will be contained in $$B_{\overline{p}}(\textbf{y}, \epsilon)$$ so that we have $$\textbf{x} \in B_d (\textbf{y}, \epsilon) \subset B_{\overline{p}}(\textbf{y}, \epsilon)$$ for each $$\textbf{x} \in B_{\overline{p}}(\textbf{y}, \epsilon)$$.

Next, we have the inclusion of the $$\ell^2$$-topology in the box topology. We assert that for every point $$\textbf{x}$$ of every basis element $$B_{\ell}$$ of the $$\ell^2$$-topology, there is a basis element $$B_b$$ of the box topology such that $$\textbf{x} \in B_b \subset B_{\ell}$$.

Let $$B_d (\textbf{y}, \epsilon)$$ be an $$\epsilon$$-ball in the $$\ell^2$$-topology centered at some $$\textbf{y} \in X$$. For any $$\textbf{x} \in B_d (\textbf{y}, \epsilon)$$, we can define the following basis element in the box topology: $$B_b = (x_1 - \delta_1, x_1 + \delta_1) \times (x_2 - \delta_2, x_2 + \delta_2) \times \cdots$$ where $$x_i$$ is the $$i$$-th coordinate of $$\textbf{x}$$ such that $$\sum \delta_i^2$$ converges and $$d(\textbf{y, x}) + \sqrt{\sum \delta_i^2} < \epsilon$$. Then $$B_b$$ is contained in $$B_d(\textbf{y}, \epsilon)$$ so that we have $$\textbf{x} \in B_b \subset B_d(\textbf{y}, \epsilon)$$. Geometrically, we can think of this as ensuring that the 'corners' of the infinite-dimensional 'rectangle' we've defined around $$\textbf{x}$$ are always inside the infinite-dimensional ball we've defined around $$\textbf{y}$$, which subsequently ensures that every point of the rectangle is in the ball.

$\tag*{\blacksquare}$