## 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 $|r|=\begin{cases}r&\text{ if }r\ge 0\\-r&\text{ if }r<0\end{cases}$

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 $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Example 1:

### 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) 1)

### 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

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