TITLE: Can C([0,1]) be written as a countable union of compact sets?
QUESTION [5 upvotes]: Let $X=C([0,1])$ be the Banach space of continuous real valued functions on $[0,1]$ (with the $\sup$-norm). 
I am wondering if $X$ can be written as a countable union of compact sets $K_1 \subset K_2\subset K_3 \dots$?

REPLY [8 votes]: No.

If $X$ is an infinite-dimensional Banach space then the closed unit ball is not compact.
Apply the Baire Category Theorem to $X = \bigcup_{n=1}^\infty K_n$ to find an open set with compact closure.
Combine 1. and 2. to derive a contradiction.