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Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator.

Question: How does it follow that the image of $S$ is separable?

Thanks for the help.

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up vote 5 down vote accepted
  • A countable unions of separable sets is separable.
  • A subset $K$ with compact closure is separable (consider the covers $(B(x,n^{-1}))_{x\in K}$).
  • $A=\bigcup_{j\geqslant 0}B\left(0,j\right)$ and $S(A)=\bigcup_{j\geqslant 0}S\left(B\left(0,j\right)\right)$.
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The space is a countable union of balls centered in zero: $$A = \bigcup_{n\in N}B(0,n).$$

The image of $B(0,n)$ is precompact, therefore, separable. Countable union of separable sets is separable.

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