Page 241 - Advanced Linear Algebra
P. 241
Real and Complex Inner Product Spaces 225
) )"b# b ) )"c# ) ~ " ) ) b # )
prove that the polarization identity
)
)
)
º"Á #» ~ ² " b#) c " c# ³
defines an inner product on . : Evaluate º "Hint Á % » b º # Á % » to show
=
that º"Á %» ~ º"Á %» and º"Á %» b º#Á %» ~ º" b #Á %» . Then complete the
proof that º"Á %» ~ º"Á%» .
21. Let be a subspace of a finite-dimensional inner product space . Prove
:
=
that each coset in =°: contains exactly one vector that is orthogonal to .
:
Extensions of Linear Functionals
22. Let be a linear functional on a subspace of a finite-dimensional inner
:
=
product space . Let ² # ³ ~ º # Á 9 » . Suppose that = i is an extension
of , that is, O : ~ . What is the relationship between the Riesz vectors 9
?
and 9
23. Let be a nonzero linear functional on a subspace of a finite-dimensional
:
inner product space = and let 2 ~ ker ² ³ . Show that if = i is an
extension of , then 9 2 ± : . Moreover, for each vector
"2 ± : there is exactly one scalar for which the linear functional
²?³ ~ º?Á "» is an extension of .
Positive Linear Functionals on s
(
A vector # ~ ² ÁÃÁ ³ in s is nonnegative also called positive) , written
for all . The vector # is strictly positive , written # , if # is
# , if
nonnegative but not . The set s of all strictly positive vectors in s is called
b
the nonnegative orthant in s À The vector is strongly positive , written
#
for all . The set s , of all strongly positive vectors in s is
# , if bb
the strongly positive orthant in s À
Let ¢ : ¦ s be a linear functional on a subspace : of s . Then is
(
nonnegative also called positive ) , written , if
# ¬ ²#³
for all #: and is strictly positive , written , if
# ¬ ²#³
for all #:À
24. Prove that a linear functional on s is positive if and only if 9 and
strictly positive if and only if 9 . If is a subspace of s is it true
:
that a linear functional on is nonnegative if and only if 9 ?
:
