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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. | ||
+ | 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. | ||
+ | |||
+ | ==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]] | ||
+ | )] | ||
+ | |||
+ | |||
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