Question

(i) Let V be a Banach space. Prove that if V* separable then V is not separable.

(ii) Give an example of separable Banach space V which has a non-separable dual space V* . 

Answer 1

i) Proving that aBanach space is separable if its dual is separable:

Let $\{f_n\}_{n=1}^{\infin}$ be a dense subset of the unit ball $B$ in $X^*$. For each $n\isin N$, pick $x_n\isin X$ such that $f_n(x_n)>\frac{1}{2}$ . Let $Y=\overline{span\{x_n\}}$ and observe that $Y$ is separable, since finite rational combinations of $\{x_n\}$ are dense in $Y$. It is now sufficient to show that $X=Y$. We proceed by contradiction. Suppose that $X\neq Y$. Then there is an $f\isin X^*$, with $||f||=1$ such that $f(x)=0$ for all $x\isin Y$ . Now choose $n$ such that $||f_n-f||<\frac{1}{4}$. Then

$|f(x_n)|=|f_n(x_n)-f_n(x_n)+f(x_n)|\geq |f_n(x_n)|-|f_n(x_n)-f(x_n)|\geq$

$\geq |f_n(x_n)|-||f_n-f||\cdot ||x_n)||>\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$

ii) Space $L^p$