Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$
I can't figure out how to solve this exercise:
My first problem is: how to control cardinality of $\overline{ \text{span}\{K\}}$? According to the definition of separable space it has to be countable. But if the field is $\mathbb{R}$ (or $\mathbb{C}$) |$ \text{span}\{K\}$| is strictly greater than |$ \mathbb{N}$| because the span contains finite linear combinations of elements of $K$ and $K$ has at least one element.
Maybe i'm not getting the point of this exercise (Real Analysis – Royden, $4$ ed pag. $262$ n. $28$) or missing some basic properties of compact sets in normed linear spaces.
Can someone help me?
Thanks in advance
Best Answer
In the one direction, to construct a compact set with dense span from a countable dense subset, note that if $(x_n)_{n\in\mathbb{N}}$ is a sequence converging to $x_\ast$, then the set $\{ x_n : n \in \mathbb{N}\} \cup \{x_\ast\}$ is compact. Construct a convergent sequence from the countable dense subset without changing the span.
For example, if $D = \{ y_n : n\in\mathbb{N}\}$ is a countable dense subset, set $x_n = \frac{1}{1+(n+1)\lVert y_n\rVert}y_n$ for $n\in\mathbb{N}$. We then have $\lVert x_n\rVert < \frac{1}{n+1}$, so $x_n \to 0$, whence $K = \{0\} \cup \{ x_n : n \in \mathbb{N}\}$ is compact, and since $y_n = (1+(n+1)\lVert y_n\rVert)y_n \in \operatorname{span} K$, it follows that $\overline{\operatorname{span} K} \supset \overline{D} = X$, so $K$ is a compact set with dense span.
In the other direction, note that a compact metric space is separable. Show that if $\operatorname{span} K$ is dense, and $D$ a countable dense subset of $K$, then $\operatorname{span}_{\mathbb{Q}[i]} D$ is also dense.
Namely, since $\mathbb{Q}[i]$ is dense in $\mathbb{C}$, it follows that $\operatorname{span} D \subset \overline{\operatorname{span}_{\mathbb{Q}[i]} D}$, and hence
$$\overline{\operatorname{span}_{\mathbb{Q}[i]} D} = \overline{\operatorname{span} D}$$
is a closed linear subspace containing $\overline{D} = K$, hence $\overline{\operatorname{span} K} = X$. But $\operatorname{span}_{\mathbb{Q}[i]} D$ is countable since $D$ and $\mathbb{Q}[i]$ are, so $X$ is separable.