8.1. STATEMENT AND PROOF OF THE THEOREM 207

Proof: Let A ∈ O and let B ∈L (X ,Y ) with ∥B∥ ≤ 12

∥∥A−1∥∥−1

. Then∥∥A−1B∥∥≤ ∥∥A−1∥∥∥B∥ ≤ 1

2

So by Lemma 8.1.4,

(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 8.1. The reason the left over material is o(B) follows from the obser-vation that o(B) is ∑

∞n=2 (−1)n (A−1B

)n A−1 and so∥∥∥∥∥ ∞

∑n=2

(−1)n (A−1B)n

A−1

∥∥∥∥∥≤ ∞

∑n=2

∥∥∥(A−1B)n

A−1∥∥∥≤ ∥∥A−1∥∥∥∥A−1∥∥2 ∥B∥2

∞

∑n=0

12n

It follows from this that we can continue taking derivatives of 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)]

= [I(A)(B1)I(A)+o(B1)] (B)[A−1−A−1B1A−1 +o(B1)

]+

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

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

and so

D2I(A)(B1)(B) = I(A)(B1)I(A)(B)I(A)+I(A)(B)I(A)(B1)I(A)

which shows I is C2 (O). Clearly we can continue in this way which shows I is in Cm (O)for all m = 1,2, · · · . ■

Here are the two fundamental results presented earlier which will make it easy to provethe implicit function theorem. First is the fundamental mean value inequality.