From this we can see that the mapping

u

(T x)(u) = A’1 (u) f (u) + K((ur ’ tr )1/r ) x(t) dtr

x ’’ T x,

0

µ

is contractive on Vκ . Thus, Banach™s ¬xed point theorem completes the

2

proof.

Besides the linear integral equation studied above, we will investigate

some nonlinear systems in the proofs of Theorem 11 (p. 52) and Lemma 11

(p. 124). The estimates needed in the proof of the lemma are more subtle,

however; so we cannot directly apply the arguments used here.

Exercises:

1. Let f (z) be holomorphic in a region G ‚ C , and for z0 ∈ G assume

f (z0 ) = 0, f (z0 ) = 0. For su¬ciently small δ > 0, show that φ(z) =

z ’ f (z)/f (z) maps the disc about z0 of radius δ into itself and is a

contraction there.

2. Investigate φ (as above) for the case of f (z0 ) = . . . = f (n’1) (z0 ) = 0,

f (n) (z0 ) = 0, n ≥ 2.

3. Under the assumptions of Proposition 26, show that if the vector f

remains bounded at the origin (in S), or even is continuous there,

then so is the solution vector x.

Appendix B

Functions with Values in Banach

Spaces

Here, we shall brie¬‚y provide some results from the theory of holomorphic

functions with values in a Banach space which are used in Chapters 4“7,

10, and 11. Those who are not interested in such a general setting may

concentrate on functions with values in C .

Throughout the book, we shall consider ¬xed, but arbitrary, Banach

spaces E and F over the complex number ¬eld. We write L(E , F) for the

space of all linear continuous operators from E into F, which again is a

Banach space, and even a Banach algebra in case E = F. Instead of L(E , C )

we write E — . If nothing else is said, we shall denote by G some ¬xed region

in the complex domain, i.e., some nonempty open and connected subset of

C , and by f some mapping f : G ’’ E (or f : G ’’ L(E , F)). We then

call f holomorphic in G if for every z0 ∈ G the limit

f (z) ’ f (z0 )

lim = f (z0 )

z ’ z0

z’z0

exists.1 We call f weakly holomorphic in G if for every continuous linear

functional φ ∈ E — the (C -valued) function φ —¦ f is holomorphic in G.

Clearly, holomorphy implies weak holomorphy, and we shall see that the

converse holds true as well. This makes it not too much of a surprise that

many results from the theory of functions of a complex variable immediately

carry over to the Banach space situation.

1 To be exact, the above quotient of di¬erences should be interpreted as multiplication

of the vector f (z) ’ f (z0 ) from the left by (z ’ z0 )’1 ; we prefer the quotient notation

in order to point out the close analogy to the scalar situation.

220 Appendix B. Functions with Values in Banach Spaces

For an arbitrary function f , one can de¬ne the integral over f along a

path, i.e., a recti¬able curve, γ in G to be the “limit” of the Riemann sums,

in case it exists. As for the scalar case, one can prove its existence for every

continuous f . Since holomorphy implies continuity, we can form integrals

of holomorphic functions over closed paths in G and raise the question

whether they always vanish. This we shall do in the next section.

B.1 Cauchy™s Theorem and its Consequences

In what follows we will assume Cauchy™s theorem for holomorphic functions

with values in C as known, and from it derive the same for E -valued

functions. Moreover, we shall derive other results for E -valued functions as

well, which the reader may or may not know for C -valued ones.

The following theorem is the key to all the results we shall obtain later:

Theorem 65 (Cauchy™s Integral Theorem and Formula)

Assume that G is simply connected and f : G ’ E is continuous. Then

the following statements are equivalent:

(a) The function f is weakly holomorphic in G.

(b) The function f is holomorphic in G.

(c) The integral of f over any closed path in G vanishes.

(d) For every positively oriented Jordan path γ and every z in the interior

region of γ we have

1 f (w)

f (z) = dw. (B.1)

w’z

2πi γ

Proof: Suppose (a) holds. For every path γ in G and every φ ∈ E — , one can

conclude from the de¬nition of the integral via Riemann sums and linearity

of φ that

φ(f (z)) dz = φ(x), x= f (z) dz. (B.2)

γ γ

The integral on the left vanishes for a closed path γ. Hence we conclude

that x = γ f (z) dz satis¬es φ(x) = 0 for every φ ∈ E — . This implies

x = 0, according to Hahn-Banach™s theorem. Thus we see that (c) follows.

Conversely, it follows from (B.2) that (c) implies (a). Denoting the right-

hand side of (B.1) by g(z), we conclude similarly that weak holomorphy

implies φ(f (z) ’ g(z)) = 0 for every z in the interior region and every

φ ∈ E — , implying (B.1). This formula then can be seen to imply holomorphy

of f , because di¬erentiation under the integral sign is justi¬ed. Thus, the

B.2 Power Series 221

proof is completed, because we observed earlier that holomorphy implies

2

weak holomorphy.

By H(G, E ) we shall denote the set of all E -valued functions that are

holomorphic in G. It is obvious from the de¬nition that H(G, E ) again is a

vector space over C . Moreover, if f ∈ H(G, E ) and T ∈ H(G, L(E , F)) (so

that z ’ T (z) f (z) is a mapping from G into the Banach space F), then

one can show easily that T f ∈ H(G, F) and

z ∈ G.

[T (z) f (z)] = T (z)f (z) + T (z) f (z),

Similarly, for f ∈ H(G, E ) and ± ∈ H(G, C ), we conclude ±f ∈ H(G, E )

and

[±(z) f (z)] = ± (z)f (z) + ±(z) f (z), z ∈ G.

Using Cauchy™s integral formula (B.1) we now prove

Theorem 66 Every f ∈ H(G, E ) is in¬nitely often di¬erentiable, and

n! f (w)

f (n) (z) = n ≥ 0,

dw, (B.3)

(w ’ z)n+1

2πi γ

for γ and z as in (B.1).

Proof: Observe that di¬erentiation under the integral in (B.1) can be jus-

2

ti¬ed, giving the desired result.

Exercises: In the following exercises, let E , F be Banach spaces and G

some region in C .

1. For f ∈ H(G, E ) and φ ∈ E — , show

d

z ∈ G.

φ(f (z)) = φ(f (z)),

dz

2. For f ∈ H(G, E ), T ∈ H(G, L(E , F)) and ± ∈ H(G, C ), prove the

above statements on T f resp. ±f .

B.2 Power Series

As in the scalar case, one can represent holomorphic functions by in¬nite

power series, as we show now:

∞

For fn ∈ E , n ≥ 0, consider the power series n=0 fn (z ’ z0 )n (note

that again we slightly abuse notation and place the scalar factor (z ’ z0 )n

to the right of the vectors fn ). De¬ne

1/n

1/ρ = lim sup fn , (B.4)

n’∞

222 Appendix B. Functions with Values in Banach Spaces

following the usual convention of 1/0 = ∞, and vice versa. For every K >

1/ρ we then have fn ¤ K n , for every n ≥ n0 . For every k < 1/ρ,

however, we have fn ≥ k n in¬nitely often. This shows that, as in the

scalar case, the vector valued power series converges absolutely in D(z0 , ρ)

and uniformly in every strictly smaller disc, while it diverges for |z| > ρ.

∞

Expanding (w ’ z)’1 = n=0 (z ’ z0 )n (w ’ z0 )’n’1 , inserting into (B.1)

and interchanging summation and integration, which is justi¬ed because

of uniform convergence (provided that z0 is in the interior region of γ and

|z ’ z0 | < inf w∈γ |w ’ z0 |), we obtain as for the scalar situation:

Proposition 27 For z0 ∈ G, let ρ > 0 be so that D(z0 , ρ) ‚ G. Then for

every f ∈ H(G, E ) we have

∞

fn (z ’ z0 )n , z ∈ D(z0 , ρ),

f (z) = (B.5)

n=0

with coe¬cients given by

f (n) (z0 ) 1 f (z)

n ≥ 0,

fn = = dz,

(z ’ z0 )n+1

n! 2πi |z’z0 |=ρ’µ

for every su¬ciently small µ > 0.

Convergent power series with common center point z0 may be added

termwise, but in general the product of two power series is unde¬ned. How-

ever, the following holds true:

Proposition 28 For f ∈ H(G, E ), ± ∈ H(G, C ) and T ∈ H(G, L(E , F))

∞ ∞

0 fn (z ’ z0 ) , ±(z) = 0 ±n (z ’ z0 ) and T (z) =

n n

assume f (z) =

∞

0 Tn (z ’ z0 ) , for |z ’ z0 | < ρ. Then T f ∈ H(G, F), ±f ∈ H(G, E ),

n

and we have

∞ n

(z ’ z0 )n , |z ’ z0 | < ρ,

T (z) f (z) = Tn’m fm

0 m=0

∞ n

(z ’ z0 )n , |z ’ z0 | < ρ.

±(z) f (z) = ±n’m fm

0 m=0

The simple proof is left to the reader. Note that if E is a Banach algebra,

i.e., a product between elements of H(G, E ) is de¬ned, then every f ∈

H(G, E ) can be identi¬ed with an element of H(G, L(E , E )). Hence it

follows from the above propositions that H(G, E ) is a di¬erential algebra.

In particular, this holds for E = C .

Let f be given by (B.5) and choose z1 ∈ D(z0 , ρ). Then the power series

representation of f about the point z1 can be seen to be given by

∞ ∞

n+m

fn+m (z1 ’ z0 )n (z ’ z1 )m ,

f (z) =

m

m=0 n=0

B.2 Power Series 223

and the series converges at least for |z ’ z1 | < ρ ’ |z1 ’ z0 | but may

converge in a larger disc. This is important in our discussion of holomorphic

continuation in the next section.

We close this section with a brief discussion of holomorphic functions

de¬ned by integrals. While the reader should be familiar with the concept

of locally uniform convergence of series and the fact that a function repre-

sented by such a series is holomorphic once each term is so, it may be not

entirely clear what we mean by locally uniform convergence of an integral:

For a < b, where a = ’∞ and/or b = +∞ may occur, let f (t, z) be an

E -valued function of a real variable t ∈ (a, b) and a complex variable z ∈ G,

and de¬ne

b

g(z) = f (t, z) dt. (B.6)

a

We then say that the above integral converges absolutely and locally uni-

formly in G, if for every z0 ∈ G we can ¬nd an µ > 0 and a function b(t),

depending upon z0 and µ, but independent of z, such that f (t, z) ¤ b(t)

b

for |z ’ z0 | ¤ µ and every t ∈ (a, b), and so that a b(t) dt exists. Then we

have the following result on holomorphy of g:

Lemma 26 Suppose that f (t, z) is continuous in t ∈ (a, b), for ¬xed z ∈ G,

and holomorphic in z ∈ G, for ¬xed t ∈ (a, b), and the integral (B.6)

converges absolutely and locally uniformly in G. Then g(z) is holomorphic

for z in G.

Proof: Observe that f (t, z) ¤ b(t) for |z’z0 | ¤ µ implies (using Cauchy™s

∞ ’n

0 fn (t) (z ’ z0 ) , with fn (t) ¤ µ

n

formula) f (t, z) = b(t). Hence

termwise integration of the series is justi¬ed for |z ’ z0 | < µ, and doing

so gives the power series representation of g(z) about the point z0 , thus

2

proving its holomorphy.

Exercises:

∞

1. Let a power series 0 fn (z ’z0 )n , with coe¬cients fn ∈ E , converge

for |z’z0 | < ρ, ρ > 0, de¬ning an analytic function f in D = D(z0 , ρ).

Assume f (zk ) = 0 for values zk = z0 in D with limk’∞ zk = z0 . Show

that then fn = 0, n ≥ 0, hence f (z) ≡ 0.