Separability of a normed space

Question

The following is an Exercise 12, page 75 of conway's Functional Analysis.

Let $\oplus_\infty X_i = \{x\in \sqcap X_i: ||x||=\sup||x(i)||<\infty\} $ where each $X_i$ is a normed space for $i\in I$. Show that $\oplus_\infty X_i$ is separable iff $I$ is finite and each $X_i$ is separable.

My question: I can not show that when $\oplus_\infty X_i$ is separable then $I$ is finite.

thanke for your help

Answer 1

My question: I can not show that when $\oplus_\infty X_i$ is separable then $I$ is finite.

Literally, that need not be true. The set of $i$ such that $X_i \neq \{0\}$ must be finite.

Now, if $J = \left\{i \in I : X_i \neq \{0\}\right\}$ is infinite, then take a sequence $(j_n)_{n\in\mathbb{N}}$ of distinct elements of $J$, for each $j_n$ choose an $x_n\in X_{j_n}$ with $\lVert x_n\rVert = 1$, and you have an isometric embedding of $\ell^\infty(\mathbb{N})$ into $\oplus_\infty X_i$.

Thus the problem is reduced to showing that $\ell^\infty(\mathbb{N})$ is not separable.

If you don't know that (or how to do that) yet, consider that

• a separable metric space cannot contain an uncountable discrete subset, and
• $\mathcal{P}(\mathbb{N})$ is uncountable.