Problem on seperable banach spaces

1. Jul 19, 2009

logarithmic

Prove that if a Banach space X, has separable dual X*, then X is separable.

It gives the hint that the first line of the proof should be to take a countable dense subset $$\{f_n\}$$ of X* and choose $$x_n\in X$$ such that for each n, we have $$||x_n||=1$$ and $$|f_n(x)|\geq(1/2)||f_n||$$.

Ok so what do I do now. We want to show that X is separable, so it's countable dense subset would be $$\{x_n\}$$, which we just have to show is dense in X, how do I do this?

2. Jul 19, 2009

g_edgar

Those $$x_n$$ won't be dense (they all have norm 1). Instead show their linear span is dense, then argue that $$X$$ is separable.

3. Jul 19, 2009

HallsofIvy

Given any x in X, there exist a corresponding x* in X*. Since {x*} is dense in X*, for any $\epsilon> 0$, there exist a x* such that ||X*- x*||< $\epsilon$. Let x* be the member of X corresponding to x*.