Page 238 - Advanced Linear Algebra
P. 238
222 Advanced Linear Algebra
and
² #³ ~ #
for all d . A conjugate isomorphism is a bijective conjugate linear map.
If %= , then the inner product function º h Á %»¢ = ¦- defined by
ºhÁ %»# ~ º#Á %»
=
is a linear functional on . Thus, the linear map ¢ = ¦ = i defined by
%~º h Á %»
is conjugate linear. Moreover, since ºhÁ %» ~ ºhÁ &» implies % ~ & , it follows
that is injective and therefore a conjugate isomorphism (since = is finite-
dimensional).
(
Theorem 9.18 The Riesz representation theorem) Let = be a finite-
dimensional inner product space.
)
1 The map ¢= ¦ = i defined by
%~º h Á %»
is a conjugate isomorphism. In particular, for each = i , there exists a
unique vector %= for which ~º h Á %» , that is,
# ~ º#Á %»
%
for all #= . We call the Riesz vector for and denote it by 9 .
i
)
2 The map 9¢ = ¦ = defined by
9 ~ 9
is also a conjugate isomorphism, being the inverse of . We will call this
map the Riesz map .
Proof. Here is the usual proof that is surjective. If ~ , then 9 ~ , so let
us assume that £ . Then 2 ~ ker ² ³ has codimension and so
=~ º$» p 2
for $2 . Letting %~ $ for - , we require that
²#³ ~ º#Á $»
and since this clearly holds for any #2 , it is sufficient to show that it holds
for #~$ , that is,
²$³ ~ º$Á $» ~ º$Á $»
Thus, ~ ²$³° $)) and
