Suppose \(X\) is a topological space and \(F\in \{\mathbb{R},\mathbb{C}\}\). We denote the space of bounded functions from \(X\) to \(F\) by \(l^\infty(X)\). We endow \(l^\infty(X)\) with the uniform norm (or sup norm) \(\lVert f\rVert_{\infty}=\sup_{x\in X}\lvert f(x)\rvert\).
Remarks
- \(\lVert \cdot\rVert_\infty\) is indeed a norm.
- \((l^\infty(X),\lVert \cdot\rVert_\infty)\) is a Banach space.