Proposition 4

— Let E and F be two locally convex spaces and f : E \to F a continuous mapping. Assume that f(0) = 0 and that there exists a real number m \geq 0 such that f(\lambda x) = \lambda^m f(x) for every \lambda > 0. Then, if A is a bounded subset of E, f(A) is bounded in F.

In fact, if V is a neighbourhood of 0 in F, then $f^{-1}(V)$ is a neighbourhood of 0 in E. If A is bounded in E, there exists \lambda > 0 such that A \subset \lambda f^{-1}(V) and this implies that $f(A) \subset \lambda^m V$.