Is the linear span of a compact set in $L^2(X)$ complete?
More specifically, let $X$ be a locally compact Hausdorff space, and $E$ be a compact subset in $(L^2(X), ||.||_2)$. Then can we conclude that linear span of $E$, i.e, $span(E)$ is a Banach space?
My intuition says yes, but having difficulties in proving it.