# Linear functional on the set of bounded functions

Let $S$ is non-empty set, set $$l^\infty(S)=\{f:S\rightarrow\mathbb{R}: \|f\|_\infty =:\sup_{x\in S} |f(x)|<\infty\}.$$ Suppose that $\psi:l^\infty(S)\rightarrow\mathbb{R}$ is a bounded linear functional i.e. $$\psi(\alpha f+g)=\alpha\psi( f)+\psi(g),\qquad (f,g\in l^\infty(S), \alpha\in\mathbb{R})$$ and $$\huge\sup_{f\in l^\infty(S),\|f\|_\infty\leq1}|\psi(f)|<\infty\qquad$$ Prove that for any $g\in l^\infty(S)$ with $g\geq0$, $$\sup\{|\psi(f)|: 0\leq f\leq g\}=|\psi(g)|$$

• This question appears to be off-topic because it is a homework question posted verbatim. Mar 17 '14 at 15:49
• It is not homework! I need it to prove it but I don't know how.I only know it was solved in lattice theory.I need it's proof without using lattice theory. Mar 17 '14 at 15:54