Math 6210 • Real Analysis

Let \(X\) be a compact Hausdorff space. Let \(C(X)\) be the space of complex valued continuous functions on \(X\). We assume that \(C(X)\) is equipped with a norm \(\|\cdot \|\), which turns it into a Banach space. Assume moreover, that, for each \(x\in X\) the linear functional \(\lambda _x\), defined by: \(\lambda _x(f)=f(x)\) is bounded on this Banach space. Show that there exist positive constants \(A\) and \(B\), such that for every \(f\in C(X)\) \[ A\|f\|_\infty \le \|f\|\le B\|f\|_\infty . \] Here \(\|f\|_\infty \) denotes, as usual, the sup norm.