Page 98 - Advanced Linear Algebra

82 Advanced Linear Algebra

% ~ ² ÁÃÁ ³  s , we have

² ³  4
( ( ²%³ ~ c ² ³  c ((( ( ((

%

Now, if (( °4 , then ((  ( ² °4 and so % ( ³  , which implies that
is continuous at %~ .
According to the Riesz representation theorem (Theorem 9.18) and the Cauchy–
Schwarz inequality, we have
)
) ) ²%³  H ) ) % )
where 9 s . Hence, %¦ implies ²% ³ ¦ and so by linearity, % ¦ %

implies ²% ³ ¦ % and so is continuous at all .
%

Theorem 2.27 Let be a real vector space of dimension . There is a unique

=
topology on , called the natural topology , for which is a topological vector
=
=
space and for which all linear functionals on are continuous. This topology is
=
determined by the fact that the coordinate map s¢= ¦ is a
homeomorphism, where s has the standard topology induced by the Euclidean
metric.
Linear Operators on = d
A linear operator on a real vector space can be extended to a linear operator

=
d
on the complexification = d by defining

d ²" b # ³ ~ ²"³ b ²#³
Here are the basic properties of this complexification of .

Theorem 2.28 If  Á B = ² ³ , then
)
d

1 ² ³ ~ d ,  s
d
2) ²b ³ ~ d b d

d
3) ² ³ d ~ d
)
4 ´#µ ~ d d ²# d . ³
Let us recall that for any ordered basis for and any vector  # = we have
=
8
´# b µ cpx²³8 ~ ´#µ 8
Now, if is an ordered basis for , then the th column of = ´ µ 8 is
8

´ µ ~ ´ b µ cpx 8 ~ ´ ²³ d ² b ³µ cpx ²³
8
8

which is the th column of the coordinate matrix of d with respect to the basis

cpx²³. Thus we have the following theorem.
8