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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.
		+	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]]
		+	)]
		+
		+