Page 492 - Advanced Linear Algebra

476 Advanced Linear Algebra

)
Proof. We prove only part 3 . Let
B B

²!³ ~ ! and ²!³ ~ !
~ ~ ! !
Then
B
²!³ ²!³ ~ 4 4 5 ! 5 c
!
~ ~
and applying both sides of this as linear functionals to % gives
)
²

º ²!³ ²!³ “ % » ~ 45
c

~
)

The result now follows from the fact that part 1 implies ~ º ²!³ “ % » and

~ c º ² ! ³ “ % c » .
We can now present our first “umbral” result.
Theorem 19.3 For any ²!³  < and ²%³  F ,
º ²!³ “ % ²%³» ~ ºC ²!³ “ ²%³»
!
Proof. By linearity, we need only establish this for ²%³ ~ % . But if
B
²!³ ~ !
~ !
then
B

ºC ²!³ “ % » ~ L ! ² c ³! ! c % c M
~
B

~ c Á
² c ³!
~
~ b
~º ²!³ “% b »
Let us consider a few examples of important linear functionals and their power
series representations.

Example 19.2
1 We have already encountered the evaluation functional ! , satisfying
)
!
º “ ²%³» ~ ² ³