Numerical Methods in Computational Finance. Daniel J. Duffy

      Axiom A1 states that addition is associative, and axiom A2 states that addition is commutative. The element 0 is called the zero element of the vector space.

      Scalar multiplication is defined by the axioms :

      (4.3)

From these axioms we see that subtraction of vectors is possible because .

      The prototypical examples of vector spaces are n-dimensional vectors and rectangular matrices over a field K:

      (4.4)

      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 called a metric that satisfies for :