Definitions
1.NounThat which is normal or typical."Unemployment is the norm in this part of the country."
2.NounA rule that is imposed by regulations and/or socially enforced by members of a community."Not eating your children is just one of those societal norms."
3.NounA sentence with non-descriptive meaning, such as a command, permission, or prohibition.
4.NounA function which satisfies a particular set of formal conditions, created to generalize the notion of the length of a vector. Formally, a real-valued function on a vector space, generally denoted
v
↦
|
v
|
{\displaystyle v\mapsto \left|v\right|}
or
v
↦
‖
v
‖
{\displaystyle v\mapsto \left\|v\right\|}
, that satisfies the following properties:
if
v
≠
0
{\displaystyle v\neq 0}
then
‖
v
‖
≠
0
{\displaystyle \left\|v\right\|\neq 0}
;
given a scalar
k
{\displaystyle k}
,
‖
k
v
‖
=
|
k
|
⋅
‖
v
‖
{\displaystyle \left\|kv\right\|=\left|k\right|\cdot \left\|v\right\|}
, where
|
k
|
{\displaystyle \left|k\right|}
is the absolute value of
k
{\displaystyle k}
;
given two vectors
v
,
w
{\displaystyle v,w}
,
‖
v
+
w
‖
≤
‖
v
‖
+
‖
w
‖
{\displaystyle \left\|v+w\right\|\leq \left\|v\right\|+\left\|w\right\|}
(the triangle inequality).
5.Nounif
v
≠
0
{\displaystyle v\neq 0}
then
‖
v
‖
≠
0
{\displaystyle \left\|v\right\|\neq 0}
;
6.Noungiven a scalar
k
{\displaystyle k}
,
‖
k
v
‖
=
|
k
|
⋅
‖
v
‖
{\displaystyle \left\|kv\right\|=\left|k\right|\cdot \left\|v\right\|}
, where
|
k
|
{\displaystyle \left|k\right|}
is the absolute value of
k
{\displaystyle k}
;
7.Noungiven two vectors
v
,
w
{\displaystyle v,w}
,
‖
v
+
w
‖
≤
‖
v
‖
+
‖
w
‖
{\displaystyle \left\|v+w\right\|\leq \left\|v\right\|+\left\|w\right\|}
(the triangle inequality).
8.NounAny of several generalizations of the above: a field norm, ideal norm, etc.
An element of the image of some (generalized) norm, the element then said to be from the norm in question, or from the structure which gave rise to the norm."A quaternion algebra
(
a
,
b
)
{\displaystyle (a,b)}
over
k
{\displaystyle k}
splits if and only if
b
{\displaystyle b}
is a norm from the field extension
k
(
a
)
/
k
,
{\displaystyle k({\sqrt {a}})/k,}
i.e. if and only if there is some
x
{\displaystyle x}
in
k
(
a
)
{\displaystyle k({\sqrt {a}})}
which has field norm exactly equal to
b
{\displaystyle b}
."
9.NounAn element of the image of some (generalized) norm, the element then said to be from the norm in question, or from the structure which gave rise to the norm."A quaternion algebra
(
a
,
b
)
{\displaystyle (a,b)}
over
k
{\displaystyle k}
splits if and only if
b
{\displaystyle b}
is a norm from the field extension
k
(
a
)
/
k
,
{\displaystyle k({\sqrt {a}})/k,}
i.e. if and only if there is some
x
{\displaystyle x}
in
k
(
a
)
{\displaystyle k({\sqrt {a}})}
which has field norm exactly equal to
b
{\displaystyle b}
."
10.NounA high level of performance in a chess tournament, several of which are required for a player to receive a title.
11.VerbTo endow (a vector space, etc.) with a norm.