Proposition 3

— Let A be a subset of a locally convex space E. Suppose that A is bounded; then for every sequence (xn) of points of A and for every sequence (\lambda_n) of scalars tending to 0, the sequence (\lambda_n x_n) tends to 0. Conversely, if there exists a sequence (\lambda_n) of non-zero scalars such that for every sequence (x_n) of points of A, the sequence (\lambda_n x_n) is bounded, then A is bounded.*

Suppose that A is bounded. If (\lambda_n) is a sequence of scalars tending to 0, and V is a neighbourhood of 0, we have \lambda_n A \subset V whenever n is large enough, and the first assertion follows.

Conversely, if A is not bounded and if (\lambda_n) is a sequence of scalars \neq 0, then there exists a continuous semi-norm p and a sequence (x_n) of points of A, such that $p(x_n) \geq \frac{n}{|\lambda_n|}$. We have then that $p(\lambda_n x_n) \geq n$, and the sequence (\lambda_n x_n) is not bounded.