- #1
logarithmic
- 107
- 0
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 [tex]\{f_n\}[/tex] of X* and choose [tex]x_n\in X[/tex] such that for each n, we have [tex]||x_n||=1[/tex] and [tex]|f_n(x)|\geq(1/2)||f_n||[/tex].
Ok so what do I do now. We want to show that X is separable, so it's countable dense subset would be [tex]\{x_n\}[/tex], which we just have to show is dense in X, how do I do this?
It gives the hint that the first line of the proof should be to take a countable dense subset [tex]\{f_n\}[/tex] of X* and choose [tex]x_n\in X[/tex] such that for each n, we have [tex]||x_n||=1[/tex] and [tex]|f_n(x)|\geq(1/2)||f_n||[/tex].
Ok so what do I do now. We want to show that X is separable, so it's countable dense subset would be [tex]\{x_n\}[/tex], which we just have to show is dense in X, how do I do this?