Given a Hilbert space $H$ and let $K$ be a compact set in $H$. Let $X$ be the smallest closed subspace containing $K$. Prove that $X$ is separable.

Compact implies that $K$ is totally bounded. But how to use this prove $X$ is separable? And how to make use of the smallest closed subspace?

  • 1
    $\begingroup$ Hint: A compact metric space is separable. $\endgroup$ – Math1000 Dec 9 '18 at 0:13
  • $\begingroup$ @Math1000 So it suffices to prove that $X$ is compact? would you mind explain a bit more how to prove that? It is not so obvious to me. $\endgroup$ – whereamI Dec 9 '18 at 0:59
  • $\begingroup$ $X$ won't be compact (it is unbounded). @Math1000 is referring to $K$ being the compact metric space. $\endgroup$ – user25959 Dec 9 '18 at 1:18
  • $\begingroup$ Reference for the fact that $K$ is separable here: math.stackexchange.com/questions/974233/… $\endgroup$ – user25959 Dec 9 '18 at 1:18

Once you have proven the hint in the comments, let $(x_1,x_2,\ldots)$ be a countable dense sequence of $K$. Let $Q$ denote the rationals (or if you are working over $\mathbb{C}$, the set of complex numbers of the form $p+qi$ for $p,q$ rational). Observe that the set $Qx_1 := \{qx_1: q\in Q\}$ is countable. As is $Qx_1+Qx_2 := \{u+v:u\in Qx_1, v\in Qx_2\}$, and more generally, the set $C:=$ finite $Q$-linear combinations of $\{x_i\}$ is countable (this follows because a countable union of countable sets is countable).

Side claim: $X = $ closure of set of all finite linear combinations of elements of $K$, i.e. $X=\overline{\{x \in H: x=\sum_{i=1}^{N} \alpha_i k_i \text{ for some } N\in \mathbb{N},\alpha_i\in \mathbb{F},k_i\in K\}}$ (this is an easy proof).

Now what remains is to show that $C$ is dense in $X$. You can do this by showing that within $\varepsilon/2$ of any $v\in X$, there is some $u=$ a finite $\mathbb{F}$-linear combination of elements of $K$; and within $\varepsilon/2$ of $u$ there is an element $c$ of $C$.

  • $\begingroup$ Thanks a lot! I forget that the smallest closed subspace can be written as the closure of linear combinations of elements of $K$. $\endgroup$ – whereamI Dec 9 '18 at 1:22

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