Page 285 - Advanced Linear Algebra
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Metric Vector Spaces: The Theory of Bilinear Forms 269
%~º h Á %»
is an isomorphism from to = i . It follows that for each = i there exists a
=
unique vector %= for which
# ~ º#Á %»
for all #= .
The requirement that be nonsingular is necessary. As a simple example, if =
=
is totally singular, then no nonzero linear functional could possibly be
represented by an inner product.
The Riesz representation theorem applies to nonsingular metric vector spaces.
However, we can also achieve something useful for singular subspaces of a
:
nonsingular metric vector space. The reason is that any linear functional : i
can be extended to a linear functional on , where it has a Riesz vector, that
=
is,
# ~º#Á 9 » ~º h Á 9 »#
Hence, also has this form, where its “Riesz vector” is an element of , but is
=
not necessarily in .
:
(
Theorem 11.6 The Riesz representation theorem for subspaces) Let be a
:
subspace of a metric vector space . If either or is nonsingular, the linear
=
:
=
map ¢= ¦ : i defined by
%~º h Á %»O :
is surjective and has kernel : . Hence, for any linear functional : i , there
)
(
is a not necessarily unique vector %= for which ~º Á %» for all : .
Moreover, if is nonsingular, then can be taken from , in which case it is
:
:
%
unique.
Orthogonal Complements and Orthogonal Direct Sums
Definition A metric vector space = is the orthogonal direct sum of the
subspaces and , written
:
;
=~ : p ;
if =~ : l ; and : ; .
If is a subspace of a real inner product space, the projection theorem says that
:
the orthogonal complement : of is a true vector space complement of ,
:
:
that is,
=~ : p :