The tag line basically says it all...this is a question in Luenberger's Optimization book (5.14.4 on p.138). Clearly I don't expect someone to deliver a full proof if it's tedious, but a sketch or theorem or tactic would help...I tried:
characterizing the dual of $BV[a,b].$
finding a sequence $\{f_n\}$ of continuous functions without a weakly convergent subsequence.
Using the Hahn-Banach theorem to extend a functional on $BV[a,b]$.
Any help? Thanks!
The dual of $C = C[a,b]$ is the Banach space $\mathcal {M}$ of finite Borel measures on $[a,b].$ Now $C$ is separable (polynomials with rational coefficients), but $\mathcal {M}$ is not (consider $\delta_{t}, t \in [a,b]$). There's a theorem somewhere that says that if $Y$ is nonseparable, then so is $Y^*.$ If we use this theorem, we're done. But we can show $\mathcal {M}^*$ is nonseparable without this theorem: For $t\in [a,b],$ define $\varphi_t(\mu) = \int_{[a,t]}d\mu.$ Then each $\varphi_t \in \mathcal {M} ^*.$ If $a\le s < t \le b,$ then $$\|\varphi_t-\varphi_s\| \ge |(\varphi_t-\varphi_s)(\delta_{(s+t)/2)})| = 1.$$ Thus there are uncountably many elements of $\mathcal {M} ^*$ that are at distance at least $1$ from each other. This implies $\mathcal {M} ^*$ is nonseparable.
