Normed vector spaces

http://mathcs.chapman.edu/structuresold/files/Normed_vector_spaces.pdf

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\begin{document}
\textbf{\Large Normed vector spaces}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Normed_vector_spaces}{edit}

\abbreviation{NFVec}

\begin{definition}
A \emph{normed vector space} is a structure $\mathbf{A}=\langle V,+,-,\mathbf 0,s_r(r\in F),||\cdot||\rangle$ over an \href{Ordered fields.pdf}{ordered field} $\mathbf F=\langle F,+,-,0,\cdot,1,\le\rangle$ such that

$\langle V,+,-,0,s_r(r\in F)\rangle$ is a \href{Vector_spaces}{vector space} over $\mathbf F$

$||\cdot||:V\to [0,\infty)$ is a \emph{norm}:  $||x||=0\iff x=\mathbf 0$

$||rx||=|r|\cdot||x||$

$||x+y|| \le ||x||+||y||$

Remark: $rx=s_r(x)$ is the scaler product, and $|r|=\begin{cases}r&\text{ if }r\ge 0\\-r&\text{ if }r<0\end{cases}$

This is a template.
If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be normed vector spaces. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
norm-nonincreasing homomorphism: 
$h(x + y)=h(x) + h(y)$,
$h(rx)=rh(x)$,
$||h(x)||\le||x||$.
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
  Classtype                       & (value, see description) \cite{Lastname19xx} \\\hline
  Equational theory               & \\\hline
  Quasiequational theory          & \\\hline
  First-order theory              & \\\hline
  Locally finite                  & \\\hline
  Residual size                   & \\\hline
  Congruence distributive         & \\\hline
  Congruence modular              & \\\hline
  Congruence $n$-permutable       & \\\hline
  Congruence regular              & \\\hline
  Congruence uniform              & \\\hline
  Congruence extension property   & \\\hline
  Definable principal congruences & \\\hline
  Equationally def. pr. cong.     & \\\hline
  Amalgamation property           & \\\hline
  Strong amalgamation property    & \\\hline
  Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
  f(1)= &1\\
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\end{finite_members}

\begin{subclasses}\ 

  \href{Banach_spaces.pdf}{Banach spaces}

\end{subclasses}

\begin{superclasses}\ 

  \href{Metric_spaces.pdf}{Metric spaces} reduced type

  \href{Vector_spaces.pdf}{Vector spaces} reduced type

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Lastname19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview} 

\end{thebibliography}

\end{document}
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