730 CHAPTER 22. THE DERIVATIVE

Then O is open and I is in Cm (O) for all m = 1,2, · · · . Also

DI(A)(B) =−I(A)(B)I(A). (22.10.19)

In particular, I is continuous.

Proof: Let A ∈ O and let B ∈L (X ,Y ) with

||B|| ≤ 12

∣∣∣∣A−1∣∣∣∣−1.

Then ∣∣∣∣A−1B∣∣∣∣≤ ∣∣∣∣A−1∣∣∣∣ ||B|| ≤ 1

2and so by Lemma 22.10.3, (

I +A−1B)−1 ∈L (X ,X) .

It follows that

(A+B)−1 =(A(I +A−1B

))−1=(I +A−1B

)−1A−1 ∈L (Y,X) .

Thus O is an open set.Thus

(A+B)−1 =(I +A−1B

)−1A−1 =

∞

∑n=0

(−1)n (A−1B)n

A−1

=[I−A−1B+o(B)

]A−1

which shows that O is open and, also,

I(A+B)−I(A) =∞

∑n=0

(−1)n (A−1B)n

A−1−A−1

= −A−1BA−1 +o(B)

= −I(A)(B)I(A)+o(B)

which demonstrates 22.10.19. It follows from this that we can continue taking derivativesof I. For ||B1|| small,

− [DI(A+B1)(B)−DI(A)(B)] =

I(A+B1)(B)I(A+B1)−I(A)(B)I(A)

= I(A+B1)(B)I(A+B1)−I(A)(B)I(A+B1)+

I(A)(B)I(A+B1)−I(A)(B)I(A)

= [I(A)(B1)I(A)+o(B1)] (B)I(A+B1)+

I(A)(B) [I(A)(B1)I(A)+o(B1)]