| Version 9 |
Version 8 |
| 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 $(V,\norm{.})$ where $V$ is a vector space over $\mathbb{F}$ and $\norm{.}:V\rightarrow\mathbb{R}$ is a function such that |
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 $(V,\norm{.})$ where $V$ is a vector space over $\mathbb{F}$ and $\norm{.}:V\rightarrow\mathbb{R}$ is a function such that |
| \begin{enumerate} |
\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 $\norm{v}\geq 0$ 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 $\norm{\lambda v} = \abs{\lambda}\norm{v}$ 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}) |
\item $\norm{v+w}\leq\norm{v}+\norm{w}$ for all $v,w\in V$ (the \emph{triangle inequality}) |
| \end{enumerate} |
\end{enumerate} |
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| The function $\norm{.}$ is called a \emph{norm} on $V$. |
The function $\norm{.}$ is called a \emph{norm} on $V$. |
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| Some properties of norms: |
Some properties of norms: |
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| \begin{enumerate} |
\begin{enumerate} |
| \item |
\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 $W$. |
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 $W$. |
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| \item |
\item |
| Any normed vector space $(V,\norm{.})$ is a metric space under the metric $d:V \times V \to \mathbb{R}$ given by $d(u,v)=\norm{u-v}$. This is called the \emph{metric induced by the norm $\norm{.}$}. |
Any normed vector space $(V,\norm{.})$ is a metric space under the metric $d:V \times V \to \mathbb{R}$ given by $d(u,v)=\norm{u-v}$. This is called the \emph{metric induced by the norm $\norm{.}$}. |
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| It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above. |
It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above. |
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| In this metric, the norm defines a continuous map from $V$ to $\mathbb{R}$ - this is an easy consequence of the triangle inequality. |
In this metric, the norm defines a continuous map from $V$ to $\mathbb{R}$ - this is an easy consequence of the triangle inequality. |
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\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 $v \in V$. |
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 $v \in V$. |
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| The norm is a convex function of its argument. |
The norm is a convex function of its argument. |
| \end{enumerate} |
\end{enumerate} |
