Chapter III

Spaces of continuous linear mappings

In this chapter, all the vector spaces under consideration are vector spaces over a field $\mathbf{K}$, which may be $\mathbf{R}$ or $\mathbf{C}$.

We recall (II, p. 2) that a semi-normed space is a vector space E endowed with a semi-norm $p$ and with the topology defined by $p$. Let $r$ be a real number $> 0$. The set of all $x \in E$ such that $p(x) \leq r$ is called the ball (closed) of radius $r$ of E (or of $p$). When $r = 1$, this ball is also called the unit ball.