# Differences

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normed_vector_spaces [2010/07/29 15:46] (current)
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+=====Normed vector spaces=====
+
+Abbreviation: **NFVec**
+
+====Definition====
+A \emph{normed vector space} is a structure $\mathbf{A}=\langle V,+,-,\mathbf 0,s_r(r\in F),||\cdot||\rangle$ over an [[ordered field]] $\mathbf F=\langle F,+,-,0,\cdot,1,\le\rangle$ such that
+
+$\langle V,+,-,0,s_r(r\in F)\rangle$ is a [[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.
+
+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.
+
+==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:
+$h(x + y)=h(x) + h(y)$,
+$h(rx)=rh(x)$,
+$||h(x)||\le||x||$.
+
+====Definition====
+An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +...\rangle$ such that
+
+$...$ is ...:  $axiom$
+
+$...$ is ...:  $axiom$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+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.
+
+^[[Classtype]]                        |(value, see description) [(Lastname19xx)]  |
+^[[Equational theory]]                | |
+^[[Quasiequational theory]]           | |
+^[[First-order theory]]               | |
+^[[Locally finite]]                   | |
+^[[Residual size]]                    | |
+^[[Congruence distributive]]          | |
+^[[Congruence modular]]               | |
+^[[Congruence $n$-permutable]]        | |
+^[[Congruence regular]]               | |
+^[[Congruence uniform]]               | |
+^[[Congruence extension property]]    | |
+^[[Definable principal congruences]]  | |
+^[[Equationally def. pr. cong.]]      | |
+^[[Amalgamation property]]            | |
+^[[Strong amalgamation property]]     | |
+^[[Epimorphisms are surjective]]      | |
+
+====Finite members====
+
+$\begin{array}{lr} + f(1)= &1\\ + f(2)= &\\ + f(3)= &\\ + f(4)= &\\ + f(5)= &\\ +\end{array}$
+$\begin{array}{lr} + f(6)= &\\ + f(7)= &\\ + f(8)= &\\ + f(9)= &\\ + f(10)= &\\ +\end{array}$
+
+
+====Subclasses====
+  [[Banach spaces]]
+
+
+====Superclasses====
+  [[Metric spaces]] reduced type
+
+  [[Vector spaces]] reduced type
+
+
+====References====
+
+[(Lastname19xx>
+F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
+)]
+
+