Page 510 - Advanced Linear Algebra

494 Advanced Linear Algebra

º d ² ! ³ “ % » ~
for all   and so ² d ²!³³ € . Thus, ² d ²!³³ ¦ B and d is

continuous.
;
;
For the converse, assume that is continuous. If did have the form d , then

º;!“ % » ~ º d !“ % » ~ º!“ % »

and since

º! “ % »

%~ %
[
‚
we are prompted to define by

º;! “ % »

%~ %
[
‚

This makes sense since ²;! ³ ¦ B as ¦ B and so the sum on the right is a
finite sum. Then

º;! “ % »
d

º ! “% »~º! “ % » ~ º! “ % »~º;! “% »

[
‚
which implies that ;! ~ d ! for all ‚ . Finally, since and ; d are both
continuous, we have ;~ d .
Umbral Operators and Automorphisms of the Umbral Algebra
Figure 19.1 shows the map , which is an isomorphism from the vector space

BF²³ onto the space of all continuous linear operators on . We are interested
<
in determining the images under this isomorphism of the set of umbral operators
and the set of umbral shifts, as pictured in Figure 19.1.