Page 97 - Advanced Linear Algebra
P. 97
Linear Transformations 81
6
in . But if # 6 , then there is an open interval 0 s for which 0 and
0# 6. Hence, the entire open set
< ~ s dÄd s d0 d s dÄd s
where the factor is in the th position is in c ² 6 ³ , that is,
0
² ÁÃÁ ³ < c ²6³
Thus, c is open and , and therefore also , is continuous.
²6³
Next we show that if every linear functional on = is continuous under a
topology on , then the coordinate map is continuous. If # = denote by
J
=
th coordinate of ´#µ 8Á the . The map defined by
´#µ 8 ¢ = ¦ s # ~ ´#µ 8 Á is a
linear functional and so is continuous by assumption. Hence, for any open
interval 0 s the set
( ~ ¸# = ´#µ Á 8 0 ¹
is open. Now, if are open intervals in , then
s
0
c
²0 dÄd0 ³ ~ ¸# = ´#µ 0 dÄd0 ¹ ~ (
8
is open. Thus, is continuous.
We have shown that if a topology J has the property that ²= Á ³ is a
J
topological vector space under which every linear functional is continuous, then
J
and ~ c are homeomorphisms. This means that if exists, its open sets
must be the images under of the open sets in the standard topology of s . It
remains to prove that the topology on that makes a homeomorphism
=
J
makes ²= Á ³ a topological vector space for which any linear functional on =
J
is continuous.
The addition map on is a composition
=
7 ~ c 7 k Z k ²d³
Z
where 7s d ¢ s ¦ s is addition in s and since each of the maps on the
right is continuous, so is .
7
Similarly, scalar multiplication in is
=
C ~ c C k Z k d ² ³
Z
where Cs d ¢ s ¦ s is scalar multiplication in s . Hence, C is
continuous.
Now let be a linear functional. Since is continuous if and only if k c is
continuous, we can confine attention to =~ s . In this case, if Á Ã Á is the
standard basis for s and ( ( ² ³ 4 for all , then for any
