Page 490 - Advanced Linear Algebra
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474 Advanced Linear Algebra
3
As it happens, any linear functional on has the form ² ! F ³ . To see this, we
simply note that if
B º3 % »
²!³ ~ ! !
3
~
then
º ²!³ % » ~ º3 % »
3
for all and so as linear functionals, 3 ~ ²!³ .
3
Thus, we can define a map F ¦ ¢ i < by 3 ² ³ ~ ² ! . ³ 3
Theorem 19.1 The map F ¦ ¢ i < defined by 3 ² ³ ~ ² ! ³ 3 is a vector space
isomorphism from F i onto .
<
Proof. To see that is injective, note that
²!³ ~ ²!³¬º3 % »~º4 % » for all ¬3~4
3
4
Moreover, the map is surjective, since for any < , the linear functional
3 ~ ²!³ has the property that ²3³ ~ ²!³ ~ ²!³. Finally,
3
B º 3 b 4 % »
² 3 b 4³ ~ !
~ !
B º3 % » B º4 % »
~ ! b !
! !
~ ~
~ ²3³ b ²4³
From now on, we shall identify the vector space F i with the vector space ,
<
using the isomorphism F ¦ ¢ i < . Thus, we think of linear functionals on F
simply as formal power series. The advantage of this approach is that is more
<
than just a vector space—it is an algebra. Hence, we have automatically defined
a multiplication of linear functionals, namely, the product of formal power
series. The algebra , when thought of as both the algebra of formal power
<
series and the algebra of linear functionals on , is called the umbral algebra .
F
Let us consider an example.
Example 19.1 For d , the evaluation functional F i is defined by
²%³» ~ ² ³
º
