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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.

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    $\begingroup$ This is of interest. $\endgroup$ – David Mitra May 21 '15 at 20:37
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    $\begingroup$ Note: in $\ell^1$, weak convergence of sequences is the same as norm convergence. This doesn't hold for filters or nets. $\endgroup$ – Daniel Fischer May 21 '15 at 20:37
  • $\begingroup$ I think @DavidMitra answered my question. $\endgroup$ – Peter May 21 '15 at 20:42
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The answer is "no".

The unit sphere is norm closed in the unit ball under the (norm) subspace topology.

But in an infinite dimensional normed space, the weak closure of the unit sphere is the unit ball. See this post for a proof of this.

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 $B(X, {\rm weak})$.

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