Proving Positivity of Extension of Linear Functional on $C(X)$

Question

Question

I am currently working through Royden's proof that any continuous linear operator $L$ on $C(X)$ can be written as the difference of two positive linear functionals on $C(X)$, where $X$ is a compact Hausdorff space equipped with the maximum norm.

The proof starts by showing that $L_+ : C(X) \to \mathbb{R}$ with

$$ L_+(f) := \sup_{0 \leq \psi \leq f}L(\psi)$$

is a non-negative functional when restricted to the non-negative elements of $C(X)$. To extend this definition to arbitrary $f \in C(X)$, Royden starts by choosing $M \geq 0$ such that $f + M \geq 0$ (which can be done since $f$ is bounded). He then shows that $L_+(f+M) - L_+(M)$ is independent of the choice of $M$, and accordingly defines $L_+(f)$ to be this value. He then claims that under this definition, "clearly $L_+(f):C(X) \to \mathbb{R}$ is positive." I am having a hard time showing this last fact, and would appreciate any insight into how to show this.

My work so far

My best approach has been to split $f = f^+ - f^-$. Then \begin{align} L_+(f + M) - L_+(M) &= L_+(f^+ - f^- + M) - L_+(M) \\ &= L_+(f^+) - L_+(M) + L_+(M-f^-). \end{align}

The second step follows from

1. The linearity of $L_+$ when restricted to non-negative functions on $C(X)$
2. $f^- \leq M$ since $f + M \geq 0$
3. $f^+$ and $f^-$ are in $C(X)$ and non-negative.

Now the third term on the RHS is non-negative by construction of $L_+$ thus far. However, I can't quite demonstrate whether the difference of the first and second terms is also non-negative.

Please let me know if I can provide any additional information. Thanks!

Answer 1

Positive here means " non-negative functional when restricted to the non-negative elements of $C(X)$".

A nonzero linear functional cannot be non-negative everywhere, as it would not be linear. In the concrete case of your $L_+$, if you evaluate at the constant function $f=-M$, you get $-L_+(M)$, which can only be zero if $L_+=0$.