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Metric Vector Spaces: The Theory of Bilinear Forms 265
Orthogonality
As we will see, not all metric vector spaces behave as nicely as real inner
product spaces and this necessitates the introduction of a new set of terminology
to cover various types of behavior. (The base field is the culprit, of course.)
-
The most striking differences stem from the possibility that º%Á %» ~ for a
nonzero vector %= .
The following terminology should be familiar.
Definition Let be a metric vector space. A vector is orthogonal to a vector
%
=
& %, written & º % Á & , if » ~ % . A vector = is orthogonal to a subset of
:
= % , written : º % Á » ~, if for all : : . A subset of is orthogonal to a
=
subset ; of = , written : ; , if º Á ! » ~ for all : and ! ; . The
orthogonal complement ? of a subset of is the subspace
?
=
?~ ¸# = # ?¹
Note that regardless of whether the form is symmetric or alternate and hence
(
)
skew-symmetric , orthogonality is a symmetric relation, that is, %& implies
& %. Indeed, this is precisely why we restrict attention to these two types of
bilinear forms.
There are two types of degenerate behaviors that a vector may possess: It may
be orthogonal to itself or, worse yet, it may be orthogonal to every vector in .
=
With respect to the former, we have the following terminology.
Definition Let be a metric vector space.
=
(
)
1 A nonzero % = is isotropic or null ) if º%Á %» ~ ; otherwise it is
nonisotropic.
2 = ) is isotropic if it contains at least one isotropic vector. Otherwise, is
=
)
(
nonisotropic or anisotropic .
3 = ) is totally isotropic that is, symplectic if all vectors in = ( ) are
isotropic.
Note that if is an isotropic vector, then so is # for all - . This can be
#
expressed by saying that the set of isotropic vectors in is a cone in . (A
=
=
0
cone in is a nonempty subset that is closed under scalar multiplication.)
=
With respect to the more severe forms of degeneracy, we have the following
terminology.
Definition Let be a metric vector space.
=
