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\begin{document}

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\section*{norm}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

\hypertarget{analytic_geometry}{}\paragraph*{{Analytic geometry}}\label{analytic_geometry}

[[!include analytic geometry -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{on_an_abelian_group}{On an abelian group}\dotfill \pageref*{on_an_abelian_group} \linebreak
\noindent\hyperlink{on_a_vector_space}{On a vector space}\dotfill \pageref*{on_a_vector_space} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{minkowski_functionals}{Minkowski Functionals}\dotfill \pageref*{minkowski_functionals} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{dreamUnique}{Uniqueness}\dotfill \pageref*{dreamUnique} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{on_an_abelian_group}{}\subsubsection*{{On an abelian group}}\label{on_an_abelian_group}

For $(A,+)$ an [[abelian group]], then a \emph{norm} on the group is a [[function]]

\begin{displaymath}
{\vert-\vert} \;\colon\; G \longrightarrow \mathbb{R}
\end{displaymath}
to the [[real numbers]], such that

\begin{enumerate}%
\item (positivity) $(g \neq 0) \Rightarrow (\vert g\vert \gt 0)$


\item ([[triangle inequality]]) ${\vert g + h\vert}\leq {\vert g\vert} + {\vert h\vert}$


\item (linearity) ${\vert k g\vert} = {\vert k\vert} {\vert g\vert}$ for all $k \in \mathbb{Z}$.



\end{enumerate}
Here ${\vert k\vert} \in \mathbb{N}$ denotes the [[absolute value]].

A group with a norm is a \emph{[[normed group]]}, see there for more.

\hypertarget{on_a_vector_space}{}\subsubsection*{{On a vector space}}\label{on_a_vector_space}

For $k$ a [[field]] equipped with a [[valuation]] (most usually, a [[local field]] such as $\mathbb{R}$, $\mathbb{C}$, or a [[p-adic field|p-adic]] [[complete field|completion]] of a [[number field]]), a \textbf{norm} on a $k$-[[vector space]] $V$ is a [[function]]

\begin{displaymath}
{\vert-\vert} \colon V \to \mathbb{R}
\end{displaymath}
such that for all $\lambda \in k$, $v,w \in V$ we have

\begin{enumerate}%
\item ${\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert}$ (where $\vert \lambda \vert$ denotes the [[valuation]])


\item ${\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert}$ (``[[triangle inequality]]'')


\item if ${\vert v\vert} = 0$ then $v = 0$.



\end{enumerate}
If the third property is not required, one speaks of a \textbf{seminorm}.

If the triangle identity is strengthened to

\begin{itemize}%
\item ${\vert v + w\vert } \leq max ({\vert v\vert}, {\vert w\vert})$

\end{itemize}
one speaks of a \textbf{[[non-archimedean]]} seminorm, otherwise of an [[archimedean]] one.

A vector space equipped with a norm is a \textbf{normed vector space}. A vector of norm 1 is a [[unit vector]].

Each seminorm determines a [[topology]], which is [[Hausdorff space|Hausdorff]] precisely if it is a norm.

A [[topological vector space]] is called \textbf{(semi-)normed} if its [[topology]] can be induced by a (semi-)norm.

Two seminorms ${\vert - \vert}_1$ and ${\vert - \vert}_2$ are called \textbf{equivalent} if there are $0 \lt C, C' \in \mathbb{R}$ such that for all $v$ we have

\begin{displaymath}
C {\vert v \vert}_1 \leq {\vert v \vert}_2
  \leq C' {\vert v \vert}_1
  \,.
\end{displaymath}
Equivalent seminorms determine the same [[topology]].

The collection of (bounded) multiplicative seminorms on a ([[Banach space|Banach]]) [[ring]] is called its [[analytic spectrum]] (see there for details).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item The standard [[absolute value]] is a norm on the [[real numbers]].


\item More generally, on any [[Cartesian space]] $\mathbb{R}^n$ the \textbf{Euclidean norm} is given by

\begin{displaymath}
\vert \vec x\vert
     \;\coloneqq\;
   \sqrt{
   \underoverset{i = 1}{n}{\sum}
     (x_i)^2
   }
   \,.
\end{displaymath}


\end{itemize}
\begin{enumerate}%
\item more generally, for $n \in \mathbb{N}$, and $p \in \mathbb{N}$, $p \geq 1$, then the [[Cartesian space]] $\mathbb{R}^n$ carries the \emph{[[p-norm]]}

\begin{displaymath}
{\vert \vec x \vert}_p  \coloneqq \root p {\sum_i {|x_i|^p}}
\end{displaymath}

\item The [[p-norm]] generalizes to [[sequence spaces]] and [[Lebesgue spaces]].



\end{enumerate}
\hypertarget{minkowski_functionals}{}\subsubsection*{{Minkowski Functionals}}\label{minkowski_functionals}

Let $V$ be a vector space and $B \subseteq V$ an [[absorbing]] [[absolutely convex]] subset. The \textbf{Minkowski functional} of $B$ is the function $\mu_B \colon V \to \mathbb{R}$ defined by:

\begin{displaymath}
\mu_B(v) = \inf\{t \gt 0 : v \in t B\}
\end{displaymath}
This is a semi-norm on $V$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item [[Hahn-Banach theorem]]

\end{itemize}
The (open or closed) [[unit ball]] of a seminormed vector space is a [[convex set]], a [[balanced set]] and an [[absorbing set]]. The first two of these properties make the unit ball (or even any ball of positive radius) an [[absolutely convex set]].

\hypertarget{dreamUnique}{}\subsubsection*{{Uniqueness}}\label{dreamUnique}

In [[dream mathematics]], a given real [[vector space]] (with no topological structure) can have at most one complete norm, up to topological equivalence ([[homeomorphism]] of the [[identity function]]). It can have multiple inequivalent complete seminorms and incomplete norms, but their Hausdorff quotients and completions must be different. For example, the various [[Lebesgue norms]] on a [[Cartesian space]] $\mathbb{R}^n$ for finite $n$ are complete and equivalent; on $\mathbb{R}^\infty$, they are inequivalent but incomplete.

As dream mathematics includes [[excluded middle]] and [[dependent choice]], the existence of inequivalent complete norms on a given vector space cannot be proved without a stronger form of the [[axiom of choice]], enough to disprove the [[Baire property]] (which is the only classically false axiom needed in the proof of uniqueness). In \emph{[[HAF]]}, it is argued that this explains why, in [[applied mathematics]], there tends to be only one norm considered on any particular vector space (after Hausdorff completion).

This theorem applies more generally to [[F-norms]] but not to [[G-norms]] (even on a real vector space).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[metric]]


\item [[quotient norm]]


\item [[F-norm]], [[G-norm]]


\item in [[representation theory]]: [[norm map]]



\end{itemize}
[[!include analytic geometry ingredients -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Normed_vector_space}{Normed vector space}}


\item [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], \emph{[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry}, 1984 (\href{http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf}{pdf})


\item \emph{[[HAF]]}, \S{}27.47.b (for uniqueness in dream mathematics)



\end{itemize}
[[!redirects norms]] [[!redirects seminorm]] [[!redirects seminorms]] [[!redirects semi-norm]] [[!redirects semi-norms]] [[!redirects Minkowski functional]] [[!redirects Minkowski functionals]] [[!redirects normed vector space]] [[!redirects normed vector spaces]]

[[!redirects normed vector space]] [[!redirects normed vector spaces]] [[!redirects normable vector space]] [[!redirects normable vector spaces]] [[!redirects pseudonormed vector space]] [[!redirects pseudonormed vector spaces]] [[!redirects seminormed vector space]] [[!redirects seminormed vector spaces]]

[[!redirects normed space]] [[!redirects normed spaces]] [[!redirects normable space]] [[!redirects normable spaces]] [[!redirects pseudonormed space]] [[!redirects pseudonormed spaces]] [[!redirects seminormed space]] [[!redirects seminormed spaces]]

[[!redirects vector measure]] [[!redirects vector measures]]



\end{document}