Consider the set of functions $S=\{\sin(2^nx):n\in\mathbb{N}\}$ in $L^2[-\pi,\pi]$ with the metric $d(f,g)=\left(\int_{\pi}^{\pi}|f(x)-g(x)|^2dx\right)^{\frac1{2}}$. Then is $S$ both closed and bounded in $L^2[-\pi,\pi]$ but noncompact in it?
I think yes. The proof of bounded ness seems easy. The proof of closed, though a little tough, can be somewhat accomplished, by noting that a function that converges in the given metric should also converge in the standard metric norm (I think). The proof of non compactness or compactness baffles me. What is an open cover in $L^2[-\pi,\pi]$ for $S$. How to give an open cover that does not have a finite subcover? Any hints? Thanks beforehand.
$(\frac 1 {\sqrt {2\pi}} \sin kx)$ is an orthonormal sequence. The given sequence is a subsequence of this. The distance between any two terms of this sequence is $\sqrt 2$. This proves that the sequence is closed and also that there is no convergent subsequence.