Page 488 - Advanced Linear Algebra

472 Advanced Linear Algebra

has a multiplicative c , if and only if ² inverse, denoted by ³ ~ . We
leave it to the reader to show that

² ³ ~ ² ³ b ² ³
and

² b ³ ‚ min¸ ² ³Á ² ³¹
If is a sequence in with ² ³ ¦ B as ¦ < , then for any series
B
²!³ ~ !

~
we may substitute for to get the series

!
B
²!³ ~ ²!³

~
which is well-defined since the coefficient of each power of is a finite sum. In
!

particular, if ² ³ ‚ , then ² ³ ¦ B and so the composition
B

² k ³²!³ ~ ² ²!³³ ~ ²!³

~
is well-defined. It is easy to see that ² k ³ ~ ² ³ ² ³ .

If ² ³ ~ , then has a compositional inverse, denoted by and satisfying
² k ³²!³ ~ ² k ³²!³ ~ !

A series with ² ³ ~ is called a delta series .
The sequence of powers of a delta series forms a pseudobasis for , in the
<

sense that for any  < , there exists a unique sequence of constants for
which
B

²!³ ~ ²!³

~
Finally, we note that the formal derivative of the series 19.1 is given by
)
(
B
Z
C ²!³ ~ ²!³ ~ ! c

!
~
The operator is a derivation, that is,
C !
C ² ³ ~ C ² ³ b C ² ³
!
!
!