= Jordan decomposition of a bounded functional on C(K)
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{title2=$\phi=\phi^+-\phi^-,\quad\|\phi\|=\|\phi^+\|+\|\phi^-\|$}
For a bounded real <linear functional> on the <space of continuous functions on a compact space>, define $\phi^+(f)=\sup_{0\leq g\leq f}\phi(g)$ for $f\geq0$. The decomposition $g=\min(g,f)+(g-f)^+$ for $0\leq g\leq f+h$ proves additivity on the positive cone; positive homogeneity is immediate. Extend by differences, and put $\phi^-=\phi^+-\phi$. Both are <positive linear functionals>. Their norms sum to $2\phi^+(1)-\phi(1)=\sup_{\|g\|_\infty\leq1}\phi(g)=\|\phi\|$, using the substitution $g=2h-1$.
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