A linear map between Banach spaces is bounded whenever its graph is closed.Suggest a slogan
Let \(X\) and \(Y\) be Banach spaces over the same field \(\RR\) or \(\CC\), and let \(T\colon X\to Y\) be a linear map whose graph \[ \Gamma(T)\coloneqq\theset{(x,Tx)\suchthat x\in X}\subseteq X\times Y \] is closed in \(X\times Y\) with the product topology. Then \(T\) is bounded: there exists \(C\geq0\) with \(\norm{Tx}_Y\leq C\norm{x}_X\) for all \(x\in X\).