Jordan Decomposition of Dual Space of Function Space

Question

$\textbf{Background:}$

When $X$ is a locally compact Hausdorff space, there is a Jordan decomposition theorem for the dual of $C_{0}(X,\mathbb{R})$. In particular, if $I\in C_{0}(X, \mathbb{R})^{*}$, then we may find positive linear functionals $I^{+}$ and $I^{-}$ so that $I = I^{+} - I^{-}$.

One might guess that something like this is always true. If $B$ is a Banach space consisting of real-valued functions $f:X\rightarrow\mathbb{R}$, then we may define positive functionals of $B^{*}$ to be the functionals $I\in B^{*}$ so that if $f\ge 0$ then $I(f)\ge 0$.

$\textbf{Question:}$

(1) With $B$ as above, can you always write $I\in B^{*}$ as $I^{+} - I^{-}$? Is there an illustrative counterexample?

(2) If this is false in general, is there an example of a more general theorem than the one given at the very beginning of this post?

$\textbf{Remark:}$ I believe that Decomposition of functionals on sobolev spaces might give a counterexample, but I have not studied the duality of Sobolev spaces.

Answer 1

Say $B=C^1([-1,1])$ and $If=f'(0)$.

It might be interesting to determine hypotheses on $B$ that imply every linear functional is the difference of two positive functionals.