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:PROPERTIES:
:ID: 9a1cc2d9-ef99-436c-8c21-9e68fd7df192
:END:
#+title: normed vector space
#+author: Preston Pan
#+description: Length matters!
#+options: broken-links:t
* Introduction
A normed vector space is a [[id:ab024db7-6903-48ee-98fc-b2a228709c04][vector space]] with a norm defined, which describes the "length" of the vector.
This norm obeys these properties:
\begin{align}
\label{}
\lVert ax \rVert = \lvert a \rvert \lVert x \rVert \\
\lVert x + y \rVert \le \lVert x \rVert + \lVert y \rVert
\end{align}
this gives rise to a [[id:6f24f731-60e5-4904-88d7-c63869505981][metric]] $d(x, y)$:
\begin{align}
\label{}
d(x, y) = \lVert x - y \rVert
\end{align}
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