Normed vector spaces

Abbreviation: NFVec


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}: $||x||=0\iff x=\mathbf 0$


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

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

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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.


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: $h(x + y)=h(x) + h(y)$, $h(rx)=rh(x)$, $||h(x)||\le||x||$.


An \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$


Example 1:

Basic results


Finite members


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)= &\\



[[Banach spaces]]


[[Metric spaces]] reduced type
[[Vector spaces]] reduced type


1) F. Lastname, \emph{Title}, Journal, \textbf{1}, 23–45 MRreview

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