Normed vector spaces

Abbreviation: NFVec


A 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 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 is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$


Example 1:

Basic results


Finite members

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


[[Banach spaces]]


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


1) F. Lastname, Title, Journal, 1, 23–45 MRreview