Let $C[0,1]$ be the space of continuous functions, equipped with the $\sup$-norm.
My question is, how one can prove, that this space is not locally compact? Is it possible to show this explicitly, by providing a sequence of continuous functions, which does not contain a convergent subsequence?
Best Answer
Consider the intervals $E_n=[1/n,1/(n+1)]$, $n\in\mathbb N$. Define continuous functions $f_n$ such that $f_n=0$ outside $E_n$, and $f_n=1$ on the mid point of $E_n$.
Then $\|f_n\|_\infty=1$ for all $n\in\mathbb N$, and $\|f_n-f_m\|_\infty=1$ whenever $n\ne m$, so the sequence admits no convergent subsequence.