## 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.

\[\tag*{$\blacksquare$}\]