6—Vector Spaces 1526.5 Norms
The “norm” or length of a vector is a particularly important type of function that can be defined on a vector
space. It is a function, usually denoted by‖ ‖, and that satisfies
- ‖~v‖≥ 0 ; ‖~v‖= 0if and only if~v=O~
- ‖α~v‖=|α|‖~v‖
- ‖~v 1 +~v 2 ‖≤‖~v 1 ‖+‖~v 2 ‖( the triangle inequality) The distance between two vectors~v 1 and~v 2 is taken
to be‖~v 1 −~v 2 ‖.
6.6 Scalar Product
The scalar product of two vectors is a scalar valued function oftwo vector variables. It could be denoted as
f(~u,~v), but a standard notation for it is
〈
~u,~v〉
. It must satisfy the requirements
〈
~w,(~u+~v)〉
=
〈
~w,~u〉
+
〈
~w,~v〉
2.
〈
~w,α~v〉
=α〈
~w,~v〉
3.
〈
~u,~v〉*
=
〈
~v,~u〉
4.
〈
~v,~v〉
≥ 0 ; and〈
~v,~v〉
= 0if and only if~v=O~
When a scalar product exists on a space, a norm naturally does too:‖~v‖=√〈
~v,~v〉
. (4)
That thisisa norm will follow from the Cauchy-Schwartz inequality. Not all norms come from scalar products.
Examples
Use the examples of section6.3to see what these are. The numbers here refer to the numbers of that section.
1 A norm is the usual picture of the length of the line segment. A scalar product is the usual product of lengths
times the cosine of the angle between the vectors.
〈
~u,~v〉
=~u.~v=uvcosθ. (5)4 A norm can be taken as the least upper bound of the magnitude of the function. This is distinguished from
the “maximum” in that the function may not actually achieve a maximum value; since it is bounded however,