Compactness of a subspace of $L^{2}([0,1])$ Let's consider the following space $K \subset L^{2}[0, 1]$, consisting of fucntions $x(t)$ so that $\sin{t} \leq x(t) \leq t$. How to check, whether this subspace is compact in $L^{2}[0, 1]$ or not.
I suppose that it's not, since it's relatively 'big' to be a compact space. Since, the reasonable approach is to suggest a counterexample, including sequence of functions so that it's impossible to extract convergent subsequence.
Some useful techniques, such as Arzela-Ascoli theorem, do not help much, since it's impossible to make accurate estimates from the given data.
Are there any hints that might help?

Hint: First, let $E$ be the set of $f\in L^2([0,1])$ such that $|f|\le1$. Then $E$ is not compact (hint: trig functions...)
Now, there exists a rectangle $[a,b]\times[c,d]$ contained in $\{(t,s):0\le t\le 1,\sin(t)\le s\le t\}$. Hence, given a sequence $f_n$ in $E$ with no convergent subsequence you can construct a sequence in $K$ with no convergent subsequence, by simply...
