Problem on seperable banach spaces

[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?

 

[g_edgar]

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?

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.

 

[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*.}