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Next,
which establishes the third axiom.
Finally,
Obviously, . Further, if and only if —that is, if and only if
. Thus the fourth axiom is satisfied.
Length and Distance in Inner Product Spaces
Before discussing more examples of inner products, we shall pause to explain how inner products are used to introduce notions of
length and distance in inner product spaces. Recall that in Euclidean n-space the Euclidean length of a vector
can be expressed in terms of the Euclidean inner product as
and the Euclidean distance between two arbitrary points and can be expressed as
[see Formulas 1 and 2 of Section 4.1]. Motivated by these formulas, we make the following definition.
DEFINITION
If V is an inner product space, then the norm (or length) of a vector u in V is denoted by and is defined by
The distance between two points (vectors) u and v is denoted by and is defined by
If a vector has norm 1, then we say that it is a unit vector.
EXAMPLE 3 Norm and Distance in
If and are vectors in with the Euclidean inner product, then
and
Observe that these are simply the standard formulas for the Euclidean norm and distance discussed in Section 4.1 [see Formulas 1
and 2 in that section].
