Subscript n Baseline right-parenthesis left-parenthesis y Subscript j Baseline element-of upper K comma j equals 1 comma ellipsis comma n right-parenthesis 3rd Row 1st Column Blank 2nd Column x comma y element-of upper K Superscript n Baseline 4th Row 1st Column Blank 2nd Column left-parenthesis x plus y right-parenthesis equals left-parenthesis x 1 plus y 1 comma ellipsis comma x Subscript n Baseline plus y Subscript n Baseline right-parenthesis 5th Row 1st Column Blank 2nd Column normal lamda x equals left-parenthesis normal lamda x 1 comma ellipsis comma normal lamda x Subscript n Baseline right-parenthesis comma normal lamda element-of upper K period EndLayout"/>
For matrices:
(4.5)
We now define an important non-negative real-valued function on a vector space V called a norm. It has the following properties:
(4.6)
Some examples of norms for two-dimensional vectors are:
The following norms for vectors and matrices are used in applications:
(4.7)
(4.8)
Whereas the norm is a measure of the size of a vector, it is also possible to find the distance between two vectors. A natural way to proceed is, given a set X to define a real-valued function on
A space X endowed with a metric d is called a metric space and is denoted by (X, d).
Examples of metrics are:
1
2 Let X be a non-empty set(4.9)
1 Let X be a set and let be the set of p-integrable Lebesgue functions on X. If , then a metric is:
Norms and metrics are important quantities when proving convergence results in functional and numerical analysis applications.
4.3 SUBSPACES
A non-empty subset X of a vector space
(4.10)
