Page 242 - Advanced Linear Algebra
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226 Advanced Linear Algebra
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25. Let ¢ : ¦ s be a strictly positive linear functional on a subspace of s .
Prove that has a strictly positive extension to s . Use the fact that if
< q s ~ ¸ ¹, where
b
s ~ ¸² ÁÃÁ ³ all ¹
b
and is a subspace of s , then < contains a strongly positive vector.
<
26. If is a real inner product space, then we can define an inner product on its
=
(
complexification = d as follows this is the same formula as for the ordinary
)
inner product on a complex vector space :
º" b # Á % b & » ~ º"Á %» b º#Á &» b ²º#Á %» c º"Á &»³
Show that
) ~ "
) )²" b )# ³ ) b # )
where the norm on the left is induced by the inner product on = d and the
norm on the right is induced by the inner product on .
=
