Show that the closed unit ball in $L^2[a,b]$ is not sequentially compact.

Question

I'm using Royden & Fitzpatrick's Real Analysis text to prep for an exam. This is one of the questions regarding the general properties of metric spaces.

Let $B=\{ f \in L^2[a,b] | \space ||f||_2 \le 1 \}$ be the closed unit ball in $L^2[a,b]$. Show that B fails to be compact by showing it is not sequentially compact.

I want to find a sequence of functions in this space that does not have a convergent subsequence. In the space $C[0,1]$, I have some possible solutions, but I haven't become comfortable with the $L^p$ spaces quite yet.

Answer 1

Hint: try an orthonormal sequence. This works in any infinite-dimensional Hilbert space.