Remark

If $f$ is a continuous real function, a point where $|f|$ attains its supremum is a point where one of the functions $f, -f$ attains its supremum. For a vector space $\mathscr{H}$ of continuous real functions satisfying the hypotheses of No. 3, the Props. 9 and 10 are thus trivial corollaries of Props. 5 and 7, respectively.