Page 117 - Advanced Linear Algebra
P. 117
The Isomorphism Theorems 101
i
ii
dim²= ³~ dim²= ³~ dim²= ³
it follows that is also surjective, hence an isomorphism.
Note that if dim²= ³ B , then since the dimensions of and = ii are the same,
=
we deduce immediately that = = ii . This is not the point of Theorem 3.13.
The point is that the natural map #¦# is an isomorphism. Because of this, =
is said to be algebraically reflexive . Theorem 3.13 and Theorem 3.12 together
imply that a vector space is algebraically reflexive if and only if it is finite-
dimensional.
If is finite-dimensional, it is customary to identify the double dual space = ii
=
with = and to think of the elements of = ii simply as vectors in = . Let us
consider a specific example to show how algebraic reflexivity fails in the
infinite-dimensional case.
= with basis
Example 3.4 Let be the vector space over {
~ ² ÁÃÁ Á Á Áó
where the is in the th position. Thus, = is the set of all infinite binary
sequences with a finite number of 's. Define the order ² # ³ of any # = to be
#
the largest coordinate of with value . Then ² # ³ B for all # = .
i
Consider the dual vectors , defined as usual by
(
)
i
² ³ ~ Á
For any #= , the evaluation functional has the property that
#
i
#² ³ ~ ²#³~ if ²#³
i
However, since the dual vectors i are linearly independent, there is a linear
functional = ii for which
i
² ³ ~
#
for all . Hence, does not have the form for any # = . This shows that
=
the canonical map is not surjective and so is not algebraically reflexive.
Annihilators
=
The functions = i are defined on vectors in , but we may also define on
subsets 4 of by letting
=
²4³ ~ ¸ ²#³ # 4¹
