I learned that $C[a,b]$ continuous functions on $[a,b]$ with maximum norm is separable. BUT I found it confusing to show that dual of it is not separable.. Can you give me some hint?
showing that dual space of $C[0,1]$ is not separable
1
$\begingroup$
real-analysis
1 Answers
2
Consider the Dirac measures $\delta_x$ with $x \in [a,b]$ and observe that $ \lVert \delta_x - \delta_y\rVert = \sup_{\lVert f\rVert \leq 1}\lvert f(x) - f(y)\rvert= 2$ whenever $x \neq y$ by choosing a continuous function such that $f(x) = 1$ and $f(y) = -1$.
-
0i'm not sure about what is the element of dual of C[0,1] is.. Can you explain more specifically? – 2012-10-31