Kenneth Kuttler

350 CHAPTER 14. FUNDAMENTALS

≤max{∥ f ◦ γ (t)− fn ◦ γ (t)∥ : t ∈ [a,b]}V (γ, [a,b]) .

Since the convergence is assumed to be uniform, this proves 14.13. ■Now suppose γ is continuous and bounded variation and f : γ∗×K→ C is continuous

where K is compact. Then consider the function F (w)≡∫

γf (z,w)dz

Lemma 14.4.8 The function F just defined is continuous.

Proof: The function f is uniformly continuous since it is continuous on a compact set.Therefore, there exists δ > 0 such that if |(z1,w1)− (z2,w2)|< δ , then

∥ f (z1,w1)− f (z2,w2)∥< ε

It follows that if |w1−w2|< δ , then from Theorem 14.4.7

∥F (w1)−F (w2)∥=∥∥∥∥∫

( f (z,w1)− f (z,w2))dz∥∥∥∥≤ εV (γ, [a,b])

Since ε is arbitrary, this proves the lemma. ■With this lemma, it becomes easy to give a version of Fubini’s theorem.

Theorem 14.4.9 Let γ i be continuous and bounded variation. Let f be continuouson γ∗1× γ∗2 having values in X a complex complete normed linear space. Then∫

γ1

∫γ2

f (z,w)dwdz =∫

γ2

∫γ1

f (z,w)dzdw

Proof: This follows quickly from the above lemma and the definition of the contour in-tegral. Say γ i is defined on [ai,bi]. Let a partition of [a1,b1] be denoted by {t0, t1, · · · , tn}=P1 and a partition of [a2,b2] be denoted by {s0,s1, · · · ,sm}= P2.∫

γ1

∫γ2

f (z,w)dwdz =n

∑i=1

∫γ1([ti−1,ti])

∫γ2

f (z,w)dwdz

∑i=1

∑j=1

∫γ1([ti−1,ti])

∫γ2([s j−1,s j])

f (z,w)dwdz

To save room, denote γ1 ([ti−1, ti]) by γ1i and γ2([

s j−1,s j])

by γ2 jThen if ∥Pi∥ , i = 1,2 issmall enough, Theorem 14.4.7 implies∥∥∥∥∥

∫γ1i

∫γ2 j

f (z,w)dwdz−∫

γ1i

∫γ2 j

f (γ1 (ti) ,γ2 (s j))dwdz

∥∥∥∥∥=

∥∥∥∥∥∫

γ1i

∫γ2 j

( f (z,w)− f (γ1 (ti) ,γ2 (s j)))dwdz

∥∥∥∥∥≤max

(∥∥∥∥∥∫

γ2 j

( f (z,w)− f (γ1 (ti) ,γ2 (s j)))dw

∥∥∥∥∥)

V (γ1, [ti−1, ti])

≤ εV(γ2,[s j−1,s j

])V (γ1, [ti−1, ti]) (14.14)

350 CHAPTER 14. FUNDAMENTALS< max {|| fo y(t) —froy(t)|| :t € [a,b] $V (y,[a,5]).Since the convergence is assumed to be uniform, this proves 14.13.Now suppose y is continuous and bounded variation and f : y* x K — C is continuouswhere K is compact. Then consider the function F (w) = J, f (z,w)dzLemma 14.4.8 The function F just defined is continuous.Proof: The function f is uniformly continuous since it is continuous on a compact set.Therefore, there exists 5 > 0 such that if |(z1,w1) — (z2,w2)| < 6, thenIf (zi, wi) — f (z2,W2)|| < €It follows that if |w; — w2| < 6, then from Theorem 14.4.7\|F (w1) —F (9) || = [Lo (z,w1) —f (z,wo))dzl] < eV (7, [a,])Since € is arbitrary, this proves the lemma.With this lemma, it becomes easy to give a version of Fubini’s theorem.Theorem 14.4.9 Ler y; be continuous and bounded variation. Let f be continuouson Yi X Y> having values in X a complex complete normed linear space. Then| f(zw)dwdz = | f (z,w)dzdwNYY Y24NProof: This follows quickly from the above lemma and the definition of the contour in-tegral. Say y; is defined on [a;,b;]. Let a partition of [a),b1] be denoted by {to,t1,--- jth} =P; and a partition of [a2,b2] be denoted by {50,51,-+- , 5m} = Po.I / f (zw wawde=) J Ll z,w)dwdzN “2 ([ti- ti)}) ha 18; Paw) dwdei=l jal? 1 (titiTo save room, denote ; ([f-1,ti]) by Y;; and 7 ([sj-1,5;]) by Y2;Then if ||Pj|| i= 1,2 issmall enough, Theorem 14.4.7 implies| f (z,w) dwdz— Js (% (ti) , Yo (sj)) dwdzNi Yj Ni Y2j=||/ fu f(n (4), Yo (8j))) dwdz <Ni 2 V2)max ([ (F(z) —£ (1 (ti), 72(8,)) awJ) V (1%; [4-1,4])< eV (%, [5j-1,5j]) VM, [t-1.4i) (14.14)