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
Miki
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6$\begingroup$ Since this is a common homework problem in beginning courses, I will not answer unless you identify yourself. $\endgroup$– Bill JohnsonMay 14, 2010 at 10:40
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2$\begingroup$ Why do you need a reference for this if you have a proof? I do not think it is the kind of thing you have to cite a reference for... $\endgroup$– Steven GubkinMay 14, 2010 at 15:36
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2 Answers
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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 $\varepsilon >0$.
Use the following function $\alpha: K \to R, \alpha(x)=||a+x||$, where $K=\cap^n_i ker(f_i)$.
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.
Miki
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1$\begingroup$ I rather strongly suspect that a reference might be hard to find, given that this is so very elementary and well known. $\endgroup$ May 14, 2010 at 13:47
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It's exercise V.1.10 in J. Conway, A Course in Functional Analysis, 2e, if that's any help.
answered May 14, 2010 at 14:19