TITLE: construction of a linear functional in $\mathcal{C}([0,1])$
QUESTION [6 upvotes]: Can someone help me to construct a linear functional in $\mathcal{C}([0,1])$ that does not attain its norm?
Actually, I want to prove that $\mathcal{C}([0,1])$ is not reflexive Banach space. Is it sufficient to construct that kind of functional?

REPLY [6 votes]: Knowing some deeper theorems will help to see what to do.
The Riesz representation theorem says that the continuous linear functionals on $C([0,1])$ are precisely the signed Radon measures on $[0,1]$.  It's not hard to see that any bounded measurable function is a continuous linear functional on the space of signed measures, i.e. is an element of $C([0,1])^{**}$.  So the existence of bounded discontinuous functions on $[0,1]$ shows that $C([0,1])$ cannot be reflexive.