Mathematical Structures: Normed vector spaces

# Normed vector spaces

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Changed: 131c131
 \href{Metric_spaces.pdf}{Metric spaces} supervariety
 \href{Metric_spaces.pdf}{Metric spaces} reduced type

Changed: 133c133
 \href{Vector_spaces.pdf}{Vector spaces} subreduct
 \href{Vector_spaces.pdf}{Vector spaces} reduced type

http://mathcs.chapman.edu/structuresold/files/Normed_vector_spaces.pdf
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\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
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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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

\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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Last edited July 31, 2004 9:40 pm by Jipsen (diff)
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