Page 96 - Advanced Linear Algebra
P. 96
80 Advanced Linear Algebra
then "b # ² Á ³ and so there is an for which
²" c Á " b ³ b ²# c Á # b ³ ² Á ³
for all . It follows that if
²" ÁÃÁ" ³ 0 ²" c Á " b ³ d Ä d ²" c Á " b ³ 8
and
²# Á Ã Á # ³ 1 ²# c Á # b ³ d Ä d ²# c Á # b ³ 8
then
²" ÁÃÁ" ³ b ²# ÁÃÁ# ³ 7 ²0Á1³ ² Á ³ d Ä d ² Á ³
The Natural Topology on =
Now let = be a real vector space of dimension and fix an ordered basis
8 = J ~²# Á Ã Á # ³ for . We wish to show that there is precisely one topology
on for which ² = Á J ³ is a topological vector space and all linear functionals
=
are continuous. This topology is called the natural topology on .
=
J
Our plan is to show that if ²= Á ³ is a topological vector space and if all linear
functionals on = are continuous, then the coordinate map ¢ = s 8 is a
homeomorphism. This implies that if does exist, it must be unique. Then we
J
use ~ c to move the standard topology from s to , thus giving a
=
=
8
J
J
topology for which 8 is a homeomorphism. Finally, we show that ²= Á ³ is
a topological vector space and that all linear functionals on are continuous.
=
The first step is to show that if ²= Á ³ is a topological vector space, then is
J
continuous. Since ~ where ¢ s ¦ = is defined by
² ÁÃÁ ³ ~ #
(
it is sufficient to show that these maps are continuous. The sum of continuous
)
maps is continuous. Let be an open set in . Then
J
6
C c s ²6³ ~ ¸² Á %³ d = % 6¹
is open in s d= . This implies that if % 6 , then there is an open interval
0 s containing for which
0% ~ ¸ % 0¹ 6
c
We need to show that the set ²6³ is open. But
c s ²6³ ~ ¸² Á à Á ³ # 6¹
~ s s d Äd sd¸ # 6¹ d s s d Äd
In words, an -tuple ² Á Ã Á ³ is in c ² 6 ³ if the th coordinate times is
#
