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FunctionalAnalysis

# Topological vector spaces

## Intro

### Normed spaces

A vector space $X$ is said to be a normed space if: to every $x \in X$ there is an associated non-negative $\norm{x}$, called the norm of $x$, in such a way that:

$\norm{x+y} \le \norm{x} + \norm{y}$ for all $x$ and $y$ in $X$
$\norm{\alpha x} = |\alpha| \norm{x}$ if $x \in X$
$\norm{x} \lt 0$ if $x \ne 0$.


Metric space: