Open Covers of Unit Balls in Different Spaces

Question

What are open covers of the unit ball in $C[0,1]$ (in sup norm) and $L^2[0,1]$? I'm supposed to exhibit these explicitly then show they have no finite subcover. I've seen sequential compactness arguments for $C[0,1]$ but I'm having trouble turning the sequences into a cover.

Answer 1

Let $X$ be either $C[0,1]$ or $L^{2} [0,1]$. Let $U_n =\{f \in X: \sum _{k=n}^{\infty} (\int _0 ^{1} f(x)e^{2\pi ix} dx)^{2} <1\}$. Then each $U-n$ is open in $X$ and $\{U_1,U_2,...\}$ is an open cover for the unit ball of $X$. [ I am using the fact that $\sum _{k=1}^{\infty} (\int _0 ^{1} f(x)e^{2\pi ix} dx)^{2} <\infty$ for every $f \in L^{2}[0,1]$ hence also for every $f \in C[0,1]$]. If there is a finite subcover then there exists an integer $N$ such that $\sum _{k=N}^{\infty} (\int _0 ^{1} f(x)e^{2\pi ix} dx)^{2} <1$ whenever $||f|| \leq 1$. To get a contradiction take $f(x)=e^{-2\pi iNx}$.