For linear functionals, continuity at one point, continuity everywhere, and a norm bound are equivalent.Suggest a slogan
Let \((X,\norm{\cdot})\) be a normed vector space over \(\CC\) and let \(L\colon X \to \CC\) be a linear functional. The following are equivalent:
\(L\) is continuous.
\(L\) is continuous at \(0\).
\(L\) is bounded: there exists \(c\geq 0\) such that \(\abs{L(x)} \leq c \norm{x}\) for all \(x\in X\).