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&gt; 0, there exist a x^* such that ||X^*- x^{*||< \epsilon. Let x_* be the member of X corresponding to x*.}