Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 5.3

Dependence on Initial Conditions and Parameters

Mathematics — Calculus · operator-norm-algebra

Problem

Let

X

be a Banach space. Prove that

\|AB\| \leq \|A\|\|B\|

, for every

A, B \in \mathcal{L}(X, X)

. (Therefore,

\mathcal{L}(X, X)

is an algebra.)