# Normed space where unit ball's weak and norm topology coincide?

Let $X$ be a normed space. Are there any infinite-dimensional examples such that the $(B_X, w\restriction_B) = (B_X, \|\cdot \|\restriction_B)$. In finite dimensions this obviously holds always true. I am coming up with this because in $\ell^1$ weak convergence is the same as norm convergence.

• This is of interest. – David Mitra May 21 '15 at 20:37
• Note: in $\ell^1$, weak convergence of sequences is the same as norm convergence. This doesn't hold for filters or nets. – Daniel Fischer May 21 '15 at 20:37
• I think @DavidMitra answered my question. – Peter May 21 '15 at 20:42

So, in the space $B(X, {\rm weak})$, the closure of the unit sphere is again all of $B_X$ (in general if $A$ is a subspace of a topological space $X$ and $C\subset A$, then the closure of $C$ in $A$ is the intersection of $A$ with the closure of $C$ in $X$). In particular, the unit sphere is not weakly closed in $B(X, {\rm weak})$.