Show that the closed ball in $C([0,1])$ of center $0$ and radius $1$ is not compact.
I thought it will be compact since every closed and bounded set in $\mathbb{R}$ is compact?
Why is it not compact and how can I prove it?
Real Analysis – Closed Ball is Not Compact
real-analysis
Best Answer
This answer is for posterity, and I hope someone appreciates it. $C([0,1])$ is a metric space, so it suffices to show it has a bounded sequence with no convergent subsequence. Such a sequence is $(f_n)$ where $f_n(x)=x^n$. The boundedness is obvious. The sequence converges pointwise to a noncontinuous function. No subsequence can converge (in the metric of $C([0,1])$, that is, uniformly) because if a sequence of continuous functions converges uniformly to a function, that limiting function must be continuous.