Math 6210 • Real Analysis

Consider \[ HC=\{(x_n)\in \mathbb C^{\mathbb Z}\mid |x_n|\le \frac {1}{n}\text { for all }n\in \mathbb Z\}. \] Note that \(HC\) is simply a product of (infinitely many) closed discs. As such, it is compact with respect to the product topology. The product topology on \(HC\) is defined so that sets \[ U_x(m,\delta )=\{(y_n)\in HC\mid |x_n-y_n|<\delta \text { for all }1\le n\le m\} \] provide a fundamental system of neighborhoods of \(x\). Next, note that \(HC\) can be considered a subset of the Hilbert space \(\ell ^2(\mathbb Z)\). Let \(B(x,\epsilon )\) be, as usual, the open ball of radius \(\epsilon \) in \(\ell ^2(\mathbb Z)\). Show that for every \(\epsilon >0\) there exists a positive integer \(m\) and \(\delta >0\) such that \[ U_x(m,\delta )\subseteq B(x,\epsilon )\cap HC \] and, for every positive integer \(m\) and \(\delta >0\) there exists \(\epsilon >0\) such that \[ U_x(m,\delta )\supseteq B(x,\epsilon )\cap HC \] In other words, show that the restricted topology on \(HC\) coincides with the product topology.