TITLE: Totally bounded space
QUESTION [5 upvotes]: Suppose $M=\left \{ f\in L^1([0,1])\, |\, 0<f(x)<\frac 1{\sqrt x}   \text{almost everywhere on} \, (0,1) \right\}$.
Is it true or not, that $M$ is totally bounded?

REPLY [2 votes]: Another useful set of functions in $M$ are the 'square waves'. Let $f_1$ be the indicator function of $\cup_{i=0}^{\infty} [i, i+\frac{1}{2}]$, and define $f_{n+1}(x) = f(2^n x)$. It is simple to show that $||f_n - f_m|| = \frac{1}{4}$ as long as $m\neq n$.