Math 6210 • Real Analysis

Let \(X\) be a Banach space and let \(T\) be a function which assigns to each real number \(s\in \mathbb R\) a bounded linear operator \(T(s):X\to X\). The derivative \(T'(s)\) of \(T\) at \(s\) is the operator defined by \[ T'(s)x=\lim _{h\to 0}h^{-1}(T(s+h)-T(s))x \] where the domain of \(T'(s)\) is the set of all \(x\in X\) for which this limit exists. Prove that if the domain of \(T'(s)\) is all of \(X\) for a given \(s\), then \(T'(s)\) is a bounded linear operator on \(X\).