Let \(X\) be a compact Hausdorff space and denote by \(C(X)\) the Banach space of continuous complex valued functions on \(X\) and by \(P(X)\) the cone consisting of all \(f\in C(X)\) such that \(\operatorname {Re}(f)\ge 0\). If \(H\) is a linear subspace of \(C(X)\) such that \(H\cap P(X)=(0)\), prove that
(a) \(\|\lambda +h\|_\infty \ge |\lambda |\) for all \(\lambda \in \mathbb C\), \(h\in H\) (hint: show this is equivalent to \(\|1-h\|_\infty \ge 1\) for all \(h\in H\));
(b) there exists a positive Borel measure \(\mu \) on \(X\) such that \(\mu (X)=1\) and \[ \int h\,d\mu =0 \quad \text {for all } h\in H \] (recall that a measure \(\mu \) on \(X\) is positive if \(|\mu |=\mu (X)\)).