Does a bounded set in infinite dimensional space have a finite cover of a fixed radius?

Question

Let $B$ be a bounded set in $H_0^1(\Omega)$ and $\delta<1/2$ be a constant, whether $B$ has a finite cover of radius $\delta$.

Answer 1

For the unit ball $B$ of any infinite dimensional normed space $X$ this does not hold for any $\delta<1$. Otherwise, $B\subseteq E +\delta B$ for some finite set $E$. Re-inserting this for $B$ on the right hand side, we get $B\subseteq E+\delta E + \delta^2B \subseteq L+\delta^2 B$ where $L$ is the linear hull of $E$. Inductively, $B\subseteq L+\delta^n B$ which implies $B\subseteq \overline L =L$ since finite-dimensional subspaces of normed spaces are closed. Hence the contradiction $X\subseteq L$.