normed vector spaceNormedVectorSpace2002-01-24 12:58:462006-12-08 11:03:09Definitionnormmetric induced by a normmetric induced by the norminduced norm% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\norm}[1]{\lVert #1 \rVert}
\newcommand{\abs}[1]{\lvert #1 \rvert}
\newcommand{\ip}[2]{\langle #1 , #2 \rangle}Let $\mathbb{F}$ be a field which is either $\mathbb{R}$ or $\mathbb{C}$. A \emph{\PMlinkescapetext{normed vector space}} over $\mathbb{F}$ is a pair $(V,\norm{\cdot})$ where $V$ is a vector space over $\mathbb{F}$ and $\norm{\cdot}\colon V\to\mathbb{R}$ is a function such that
\begin{enumerate}
\item $\norm{v}\geq 0$ for all $v\in V$ and $\norm{v}=0$ if and only if $v=0$ in $V$ (\emph{positive definiteness})
\item $\norm{\lambda v} = \abs{\lambda}\norm{v}$ for all $v\in V$ and all $\lambda\in\mathbb{F}$
\item $\norm{v+w}\leq\norm{v}+\norm{w}$ for all $v,w\in V$ (the \emph{triangle inequality})
\end{enumerate}
The function $\norm{\cdot}$ is called a \emph{norm} on $V$.
Some properties of norms:
\begin{enumerate}
\item
If $W$ is a subspace of $V$ then $W$ can be made into a normed space by simply restricting the norm on $V$ to $W$. This is called the induced norm on $W$.
\item
Any normed vector space $(V,\norm{\cdot})$ is a metric space under the metric $d\colon V \times V \to \mathbb{R}$ given by $d(u,v)=\norm{u-v}$. This is called the \emph{metric induced by the norm $\norm{\cdot}$}.
\item
It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above.
\item
In this metric, the norm defines a continuous map from $V$ to $\mathbb{R}$ - this is an easy consequence of the triangle inequality.
\item
If $(V, \ip{}{})$ is an inner product space, then there is a natural induced norm given by $\norm{v} = \sqrt{\ip{v}{v}}$ for all $v \in V$.
\item
The norm is a convex function of its argument.
\end{enumerate}