Walter Rudin, Functional Analysis brief theorems(Chapter General Theory) - ongoing

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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