Page 101 - Advanced Linear Algebra
P. 101
Linear Transformations 85
~c ² c ³ ~ ² c ³ b
is a projection.
)
b If and commute, then is a projection, in which case
im ² ³ ~ im ³ ² q im ³ and ²ker ³ ~ ²ker ³ b ²ker ³
²
24. Let ¢ s s ¦ be a continuous function with the property that
²% b &³ ~ ²%³ b ²&³
Prove that is a linear functional on s .
25. Prove that any linear functional ¢ s ¦ s is a continuous map.
26. Prove that any subspace of s is a closed set or, equivalently, that
:
:~ s ± : is open, that is, for any % : there is an open ball )²%Á ³
centered at with radius for which ² ) % Á ³ : .
%
27. Prove that any linear transformation ¢= ¦ > is continuous under the
natural topologies of and > .
=
28. Prove that any surjective linear transformation from to > both finite-
(
=
)
dimensional topological vector spaces under the natural topology is an
open map, that is, maps open sets to open sets.
29. Prove that any subspace of a finite-dimensional vector space = is a
:
closed set or, equivalently, that : is open, that is, for any % : there is
an open ball )²%Á ³ centered at % with radius for which
)²%Á ³ : .
30. Let be a subspace of with dim ² = ³ B .
:
=
)
a Show that the subspace topology on inherited from is the natural
:
=
topology.
)
b Show that the natural topology on =°: is the topology for which the
natural projection map ¢= ¦ = °: continuous and open.
31. If is a real vector space, then = d is a complex vector space. Thinking of
=
= d ² as a vector space = d ³ s ² s = over , show that d ³ s is isomorphic to the
external direct product = = ^ .
(
)
32. When is a complex linear map a complexification? Let be a real vector
=
space with complexification = d and let B ² = d ³ . Prove that is a
complexification, that is, has the form d for some ²= ³ if and only
B
if commutes with the conjugate map ¢= d ¦ = d defined by
²" b #³ ~ " c #.
33. Let > be a complex vector space.
a Consider replacing the scalar multiplication on > by the operation
)
²'Á $³ ¦ '$
where ' d and $ > . Show that the resulting set with the addition
defined for the vector space > and with this scalar multiplication is a
complex vector space, which we denote by > .
b Show, without using dimension arguments, that ²> ³ > ^ > .
)
d
s
