Let $\mathscr{C}[0,1]$ denote the set of continuous functions with bounded supremum and let $K=\{f\in\mathscr{C}[0,1]|\int_0^1f(t)dt=1\}$. Then is $K$ compact in the space $\mathscr{C}[0,1]$? Typically how do we characterize the compact spaces in the space of continuous functions? Will Heine-Borel property work here?
I think Heine-Borel would work, as $[0,1]$ is a compact Hausdorff space. Then, by using a function similar to spikes, or, somewhat like Dirac-Delta function, I think the space $K$ is not compact. Is my argument true? Any hints? Thanks beforehand.
Best Answer
Let $f_n(t)=(n+1)t^n$. Then $f_n \in K$ for all $n$. Can you proceed ?