Jordan decomposition of a bounded functional on C(K) ($\varphi ={\varphi }^{+}-{\varphi }^{-},\Vert \varphi \Vert =\Vert {\varphi }^{+}\Vert +\Vert {\varphi }^{-}\Vert$)

For a bounded real linear functional on the space of continuous functions on a compact space, define ${\varphi }^{+}(f)={\sup }_{0\le g\le f}\varphi (g)$ for $f\ge 0$. The decomposition $g=\min (g,f)+(g-f{)}^{+}$ for $0\le g\le f+h$ proves additivity on the positive cone; positive homogeneity is immediate. Extend by differences, and put ${\varphi }^{-}={\varphi }^{+}-\varphi$. Both are positive linear functionals. Their norms sum to $2{\varphi }^{+}(1)-\varphi (1)={\sup }_{\Vert g{\Vert }_{\infty }\le 1}\varphi (g)=\Vert \varphi \Vert$, using the substitution $g=2h-1$.