\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Needs:
Metrics ›
Norms ›
Needed by:
Complete Normed Spaces ›
Complex Distance ›
Topological Vector Spaces ›
Links:
Sheet PDF ›
Graph PDF ›

Norm Metrics

Why

If we have a norm, then we have a metric.

Motivating result

Let $(V, F)$ be a vector space. Let $f: V \to \R $ be a norm. Let $g: V \times V \to \R $ such that

\[ g(x, y) = \norm{x - y}. \]

 Then $g$ is a metric.1
  1. Future editions will include an account. ↩︎
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