Page 114 - Advanced Linear Algebra
P. 114
98 Advanced Linear Algebra
Proof. For any vector space , we have
=
i
dim²= ³ dim²= ³
=
8
since the dual vectors to a basis for are linearly independent in = i . We
have already seen that if is finite-dimensional, then dim ² = ³ ~ dim ² = i ³ . We
=
(
wish to show that if is infinite-dimensional, then dim ² = ³ dim ² = i ³ . The
=
author is indebted to Professor Richard Foote for suggesting this line of proof.)
If is a basis for and if is the base field for , then Theorem 2.7 implies
8
=
=
2
that
8
= ²2 ³
8 is the set of all functions with finite support from to and
where ²2 ³ 8 2
i
= 2 8
where 2 8 is the set of all functions from to . Thus, we can work with the
2
8
8 8 .
vector spaces ²2 ³ and 2
The plan is to show that if is a countable subfield of 2 and if is infinite,
8
-
then
2 8 3 2 8 3 8 2 8 3
dim 2 ²2 ³ dim - ~ ²- ³ dim - ²- dim 2 ³ 2
Since we may take to be the prime subfield of , this will prove the theorem.
2
-
2
-
The first equality follows from the fact that the -space 2 ² 8 ³ and the -space
²- ³ each have a basis consisting of the “standard” linear functionals
8
¸ ¹ defined by
8
# ~ Á
for all # 8 , where Á is the Kronecker delta function.
For the final inequality, suppose that ¸ ¹ - 8 is linearly independent over -
and that
~
where 2 . If ¸ ¹ is a basis for over , then ~ Á for Á -
2
-
and so
~ Á
~
Evaluating at any # 8 gives
