# Part I - General Theory

## Chapter 1 - Topological Vector Spaces

**1.2 Normed spaces** A vector space X is said to be a normed space if to every x in X there is associated a nonnegative real number llxll, called the norm of x, in such a way that

(a) $\quad|x+y| \leq|x|+|y|$ for all $x$ and $y$ in $X$

(b) $\quad|\alpha x|=|\alpha||x|$ if $x \in X$ and $\alpha$ is a scalar,

(c) $\quad|x|>0$ if $x \neq 0$

In any metric space, the open ball with center at x and radius r is the set