Page 95 - Advanced Linear Algebra

Linear Transformations 79

in which case
ker

im²c ³ ~ im² ³ q ker ² ³ and ²c ³ ~ ker ² ³ l im² ³

)
3 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
(Example 2.5 shows that the converse may be false. )
Topological Vector Spaces
This section is for readers with some familiarity with point-set topology.
The Definition
J
=
J
A pair ²= Á ³ where is a real vector space and is a topology on the set
=
= is called a topological vector space if the operations of addition
7 7¢ = d= ¦ = Á ²#Á $³ ~ # b$
and scalar multiplication
Cs d ¢ = ¦ = Á C ² Á # ³ ~ #
are continuous functions.
The Standard Topology on s
The vector space s is a topological vector space under the standard topology ,
which is the topology for which the set of open rectangles
8 's are open intervals in s ~¸0 d Ä d 0 “ 0 ¹

is a base, that is, a subset of s is open if and only if it is a union of open
rectangles. The standard topology is also the topology induced by the Euclidean
metric on s , since an open rectangle is the union of Euclidean open balls and
an open ball is the union of open rectangles.

The standard topology on s has the property that the addition function
7s d ¢ s ¦ s ¢ ² # Á $ ³ ¦ # b $
and the scalar multiplication function

Cs d ¢ s ¦ s ¢ ² Á # ³ ¦ #
are continuous and so s is a topological vector space under this topology.
Also, the linear functionals ¢ s ¦ s are continuous maps.
For example, to see that addition is continuous, if

²" ÁÃÁ" ³ b ²# ÁÃÁ# ³  ² Á ³ d Ä d ² Á ³  8