450 CHAPTER 18. LINEAR TRANSFORMATIONS

Proof: First consider the claim 18.16.

∥AB∥∞≡ sup

i, j

∣∣∣∣∣∑kAikBk j

∣∣∣∣∣≤ supi, j

∑k∥A∥

∞∥B∥

∞

= supi, j

p∥A∥∞∥B∥

∞≤ p∥A∥

∞∥B∥

∞

Now consider 18.15. From what was just shown,

∥ABn−AB∥∞= ∥A(Bn−B)∥

∞≤ p∥A∥

∞∥Bn−B∥

∞

which is assumed to converge to 0. Similarly BnA→ BA. This establishes the first part ofthe lemma. Now 18.17 follows by induction. Indeed, the result holds for k = 1. Supposetrue for n−1 for n≥ 2. Then∥∥AAn−1∥∥

∞≤ p∥A∥

∞

∥∥An−1∥∥∞≤ p∥A∥

∞pn−2 ∥A∥n−1

∞= pn−1 ∥A∥n

∞.

Consider the claim about the sum.∣∣∣∣∣∣(

m

∑k=1

Ak

)i j

∣∣∣∣∣∣=∣∣∣∣∣ m

∑k=1

(Ak)i j

∣∣∣∣∣≤ m

∑k=1∥Ak∥∞

Since this holds for arbitrary i j, it follows that∥∥∥∥∥ m

∑k=1

Ak

∥∥∥∥∥∞

≤m

∑k=1∥Ak∥∞

as claimed. The assertion 18.18 is obvious. Consider 18.19. Using the Cauchy Schwarzinequality as needed,

|Ax| ≡

∣∣∣∣∣ p

∑j=1

Ai jx j

∣∣∣∣∣≤ ∥A∥∞

p

∑j=1

∣∣x j∣∣

≤ ∥A∥∞

(p

∑j=1

12

)1/2( p

∑j=1

∣∣x j∣∣2)1/2

≤√p∥A∥∞|x|

Now consider the last claim. ∣∣Ai j +Bi j∣∣≤ ∥A∥

∞+∥B∥

∞

and so,∥A+B∥

∞≤ ∥A∥

∞+∥B∥

∞■

Thus the convention for taking the derivative above could also be obtained by

A′ (t)≡ limh→0

A(t +h)−A(t)h

because this corresponds to taking this limit for each Ai j (t).