As I remember the following is true:
Fact: for every infinite-dimensional normed space $X$ the unit sphere $S$ is weak-dense in the unit ball $B$.
Please help me find a reference.
Thanks in advance
It's exercise V.1.10 in J. Conway, A Course in Functional Analysis, 2e, if that's any help.
Right, I just want to have a reference.
As to the proof. One of them is:
Let $a \in B$. Consider a typical weak-nbd $V$ of $a$ in $X$ parameterized by the functionals $f_i \in X^*$, $i=1,2,\cdots,n$ and $\varepsilon >0$.
Since $K$ is not 0-dimensional (here we need the assumption that $X$ is infinite-dimensional) we get by intermediate value theorem that $||a+x_0||=1$ for some $x_0 \in K$. This means that $V \cap S$ is non-empty.
Thanks in advance for your information about a reference.