Assume the banach space $C([0,1], \mathbb R)$ with supremum norm. I guess that this banach space is not sigma compact, in the other word $C([0,1], \mathbb R)$ is not union of countable compact subsets of it.
My strategy : I assume it is sigma compact the because $C([0,1], \mathbb R)$ is a complete metric space then I want to use baire theorem to reach contradiction. but I am confused to complete it.
