Vector-valued holomorphic functions in several variables
aa r X i v : . [ m a t h . F A ] O c t FOLKLORE ON VECTOR-VALUED HOLOMORPHICFUNCTIONS IN SEVERAL VARIABLES
KARSTEN KRUSE
Abstract.
In the present paper we give some explicit proofs for folklore the-orems on holomorphic functions in several variables with values in a locallycomplete locally convex Hausdorff space E over C . Most of the literature onvector-valued holomorphic functions is either devoted to the case of one vari-able or to infinitely many variables whereas the case of (finitely many) severalvariables is only touched or is subject to stronger restrictions on the complete-ness of E like sequential completeness. The main tool we use is Chauchy’sintegral formula for derivatives for an E -valued holomorphic function in sev-eral variables which we derive via Pettis-integration. This allows us to gen-eralise the known integral formula, where usually a Riemann-integral is used,from sequentially complete E to locally complete E . Among the classical the-orems for holomorphic functions in several variables with values in a locallycomplete space E we prove are the identity theorem, Liouville’s theorem, Rie-mann’s removable singularities theorem and the density of the polynomials inthe E -valued polydisc algebra. Introduction
This paper is not meant as a survey article but only to ease our mind when itcomes to vector-valued holomorphic functions in several variables, i.e. holomorphicfunctions f ∶ Ω → E from an open set Ω ⊂ C d to a complex locally convex Hausdorffspace E , by giving some proofs that are missing in the literature or only touchedwith reference to the case of one variable.There is a lot of work available on C -valued holomorphic functions in severalvariables, like the books by Gunning and Rossi [22], Hörmander [23], Jarnicki andPflug [25] and Krantz [29]. But when it comes to vector-valued holomorphic func-tions then most of the work is either restricted to the case of one variable, i.e. d = ,or directly jumps to infinitely many variables, i.e. Ω is an open subset of a complexinfinite dimensional locally convex Hausdorff space F . Holomorphy of vector-valuedfunctions in infinitely many variables is discussed for instance by Mujica in [34],where F and E are Banach spaces, and by Dineen in [12], [13] for general locallyconvex spaces F and E . These references also contain results on finitely manyvariables ( F = C d ) but the emphasis is on infinitely many variables.Banach-valued holomorphic functions in one variable are handled and charac-terised by Dunford [14, Theorem 76, p. 354] and more recently by Arendt andNikolski [1], [2]. Holomorphic functions in one variable with values in a locally con-vex Hausdorff space E are considered in [27, Satz 10.11, p. 241] by Kaballo if E isquasi-complete, in [21] by Grothendieck if E has the convex compactness property(cf. [24, 16.7.2 Theorem, p. 362-363]), in [8] by Bogdanowicz if E is sequentiallycomplete, in [19], [20] by Grosse-Erdmann if E is locally complete and several equiv-alent conditions describing holomorphy are given. In particular, in all these cases Date : October 30, 2019.2010
Mathematics Subject Classification.
Primary 46E40, Secondary 32A10, 46E10.
Key words and phrases. vector-valued, holomorphic, weakly holomorphic, several variables,locally complete. holomorphy coincides with weak holomorphy which means that f ∶ Ω → E is holo-morphic if and only if the C -valued functions e ′ ○ f are holomorphic for each e ′ ∈ E ′ where E ′ is the dual space of E . Further, the interesting question is treated underwhich conditions one can replace E ′ by a separating subspace G ⊂ E ′ and still canconclude holomorphy from the holomorphy of e ′ ○ f for each e ′ ∈ G . More general,the extension problem for E -valued holomorphic functions which have weak holo-morphic extensions is studied, in one variable by Grosse-Erdmann in [20], in severalvariables by Bonet, Frerick and Jordá in [9], [17] and Vitali’s and Harnack’s typeresults are derived in [26] if E is locally complete. Further results on vector-valuedholomorpic functions in several variables may be found in [7] by Bochnak and Si-ciak where E is sequentially complete and in a survey by Barletta and Dragomir[4], extended in [3], but here E is often restricted to have the convex compactnessproperty or even to be a Fréchet space.The main purpose of the present paper is to derive some equivalent characterisa-tions of holomorphic functions in several variables with values in a locally completespace E (see Corollary 3.19, Theorem 3.20, Corollary 3.22) with explicit proofsavoiding the usual ‘like in the case of one variable’ (see e.g. the four-line [20, Sec-tion 4.1, p. 409]). Of course, the short reference to the case of one variable is oftendue to constraints, like page limits or the perception as folklore since it is knownto everyone from the field how to transfer the results from one variable to severalvariables but never written down not least because of the low chance to get it pub-lished. This is the reason why we wrote this down so that we have a reference withexplicit proofs, not more, not less. Anyway, our main tool to obtain the equiva-lent characterisations of holomorphic functions in several variables with values in alocally complete space E is Cauchy’s integral formula for derivatives which we ob-tain via Pettis-integration (Theorem 3.12). To the best of our knowledge Cauchy’sintegral formula for derivatives for holomorphic functions with values in a locallycomplete space E is not contained in the literature. Usually, Riemann-integrationis used instead of Pettis-integration and E has to be sequentially complete or thederivatives have to be considered in the completion of E . On the way to our mainTheorem 3.20 we derive Fubini’s theorem (Theorem 3.7) and Leibniz’ rule for dif-ferentiation under the integral sign (Lemma 3.11) for holomorphic functions withvalues in a locally complete space. We use our main theorem to prove some classicaltheorems like the identity theorem (Theorem 3.24), Liouville’s theorem (Theorem3.25), Riemann’s removable singularities theorem (Theorem 3.26) and the densityof the polynomials in the E -valued polydisc algebra (Corollary 3.27).2. Notation and Preliminaries
We equip the spaces R d and C d , d ∈ N , with the usual Euclidean norm ∣ ⋅ ∣ .Moreover, we denote by B r ( x ) ∶ = { w ∈ R d ∣ ∣ w − x ∣ < r } the ball around x ∈ R d withradius r > and use the same notation when R d is replaced by C d . Furthermore,for a subset M of a topological space X we denote by M the closure of M in X .For a subset M of a topological vector space X , we write acx ( M ) for the closureof the absolutely convex hull acx ( M ) of M in X .By E we always denote a non-trivial locally convex Hausdorff space ( lcHs ) overthe field K = R or C equipped with a directed fundamental system of seminorms ( p α ) α ∈ A . If E = K , then we set ( p α ) α ∈ A ∶ = {∣ ⋅ ∣} . Further, we write ̂ E for thecompletion of E and for a disk D ⊂ F , i.e. a bounded, absolutely convex set, wewrite E D ∶ = ⋃ n ∈ N nD which becomes a normed vector space if it is equipped withgauge functional of D as a norm (see [24, p. 151]). The space E is called locallycomplete if E D is a Banach space for every closed disk D ⊂ F (see [24, 10.2.1Proposition, p. 197]). In particular, every sequentially complete space is locally OLOMORPHIC 3 complete and this implication is strict. Further, we recall the following definitionsfrom [38, p. 259] and [39, 9-2-8 Definition, p. 134]. A locally convex Hausdorffspace is said to have the [metric] convex compactness property ([metric] ccp) if theclosure of the absolutely convex hull of every [metrisable] compact set is compact.Equivalently this definition can be phrased with the convex hull instead of theabsolutely convex hull. Every locally convex Hausdorff space with ccp has metricccp, every quasi-complete locally convex Hausdorff space has ccp, every sequentiallycomplete locally convex Hausdorff space has metric ccp and every locally convexHausdorff space with metric cpp is locally complete and all these implications arestrict (see [30, p. 3-4] and the references therein). For more details on the theoryof locally convex spaces see [16], [24] or [33].For k ∈ N , ∞ ∶ = N ∪ { ∞ } we denote by C k ( Ω , E ) the space of k -times continuouslypartially differentiable functions on an open set Ω ⊂ R d with values in a locallyconvex Hausdorff space E . We say that a function f ∶ Ω → E is weakly C k if e ′ ○ f ∈C k ( Ω ) ∶ = C k ( Ω , K ) for each e ′ ∈ E ′ . By L ( F, E ) we denote the space of continuouslinear operators from F to E where F and E are locally convex Hausdorff spaces.If E = K , we just write F ′ ∶ = L ( F, K ) for the dual space. We write L t ( F, E ) for thespace L ( F, E ) equipped with the locally convex topology of uniform convergenceon compact subsets of F if t = c , on the absolutely convex, compact subsets of F if t = κ and on the bounded subsets of F if t = b . The so-called ε -product of Schwartz is defined by F εE ∶ = L e ( F ′ κ , E ) where L ( F ′ κ , E ) is equipped with the topology of uniform convergence on theequicontinuous subsets of F ′ (see e.g. [37, Chap. I, §1, Définition, p. 18]). Formore information on the theory of ε -products see [24] and [27].3. Holomorphic functions in several variables
Definition ((weakly, separately, Gâteaux-) differentiable, holomorphic) . Let E be an lcHs over K , let Ω ⊂ K d be open and f ∶ Ω → E .a) f is called differentiable (on Ω ) if for every z ∈ Ω there is a K -linear map df ( z ) ∶ = d K f ( z ) ∶ K d → ̂ E such that lim w → zw ∈ Ω ,w ≠ z f ( w ) − f ( z ) − df ( z )[ w − z ]∣ w − z ∣ = in ̂ E and the map df ( ⋅ )[ v ] ∶ Ω → ̂ E is continuous for every v ∈ K d .b) f is called the Gâteaux-differentiable (on Ω ) if Df ( z )[ v ] ∶ = D K f ( z )[ v ] ∶ = lim h → h ∈ K ,h ≠ f ( z + hv ) − f ( z ) h exists in ̂ E for every z ∈ Ω and v ∈ K d .c) If v = e j is the j -th unit unit vector for ≤ j ≤ d and z ∈ Ω , we write ( ∂ e j K ) E f ( z ) ∶ = ( ∂ z j ) E f ( z ) ∶ = D K f ( z )[ e j ] if D K f ( z )[ e j ] exists in E . Especially, we use f ′ ( z ) ∶ = ( ∂ e K ) E f ( z ) if d = .d) For z = ( z , . . . , z d ) ∈ Ω we define the continuous function π z,j ∶ K → K d , π z,j ( w ) ∶ = ( z , . . . , z j − , w, z j + , . . . , z d ) .f is called separately differentiable (on Ω ) if f is a differentiable function ineach variable, i.e. f ○ π z,j ∶ π − z,j ( Ω ) → E is differentiable for every z ∈ Ω and ≤ j ≤ d .e) f is called weakly (separately, Gâteaux-) differentiable (on Ω ) if e ′ ○ f ∶ Ω → K is (separately, Gâteaux-) differentiable for every e ′ ∈ E ′ . K. KRUSE f) If K = C , we say holomorphic or complex differentiable instead of differen-tiable on the open set Ω and, if K = R , we sometimes say real differentiable .3.2. Remark.
Let E be an lcHs over K , Ω ⊂ K d open and f ∶ Ω → E .a) If f is differentiable, then df ∶ Ω × K d → ̂ E is continuous.b) If f is differentiable, then f ∶ Ω → E is continuous.c) If f is differentiable, then f is Gâteaux- and separately differentiable and df ( z )[ v ] = Df ( z )[ v ] = d ∑ j = ( ∂ e j K ) ̂ E f ( z ) v j , z ∈ Ω , v = ( v , . . . , v d ) ∈ K d . d) If f is (separately, Gâteaux-) differentiable, then f is weakly (separately,Gâteaux-) differentiable.e) If ( ∂ e j K ) ̂ E f ( z ) ∈ E for some ≤ j ≤ d and z ∈ Ω , then ( ∂ e j K ) ̂ E f ( z ) = ( ∂ e j K ) E f ( z ) . Proof. a) First, we remark that df ( z ) ∶ K d → ̂ E is continuous for every z ∈ Ω since df ( z ) is linear and K d a finite dimensional normed space. Let ( z, v ) ∈ Ω × K d , ε > and α ∈ ̂ A where ( ̂ E, ( p α ) α ∈̂ A ) is the completion of E . For every ( w, x ) ∈ Ω × K d weestimate p α ( df ( w )[ x ] − df ( z )[ v ]) ≤ p α ( df ( w )[ x − v ]) + p α ( df ( w )[ v ] − df ( z )[ v ]) ≤ √ d sup ≤ j ≤ d p α ( df ( w )[ e j ])∣ x − v ∣ + p α ( df ( w )[ v ] − df ( z )[ v ]) . Since df ( ⋅ )[ v ] ∶ Ω → ̂ E is continuous, there is δ = δ α,z,v > such that for all w ∈ Ω with ∣ w − z ∣ < δ we have p α ( df ( w )[ v ] − df ( z )[ v ]) < ε / . As Ω is open, there is δ > such that K z ∶ = B δ ( z ) ⊂ Ω . From the compactness of K z and the continuity of df ( ⋅ )[ e j ] ∶ Ω → ̂ E for every ≤ j ≤ d we deduce that C j,z ∶ = sup w ∈ K z p α ( df ( w )[ e j ]) < ∞ . Thus we obtain for every ( w, x ) ∈ Ω × K d with ∣( w, x ) − ( z, v )∣ < min ( δ, δ , ε ( + √ d sup ≤ j ≤ d C j,z ) ) that p α ( df ( w )[ x ] − df ( z )[ v ]) ≤ √ d sup ≤ j ≤ d C j,z ∣ x − v ∣ + ( ε / ) < ( ε / ) + ( ε / ) = ε. b) Let z ∈ Ω , ε > and α ∈ ̂ A . Then there is δ > such that for all w ∈ K d with < ∣ w − z ∣ < δ we have p α ( f ( w ) − f ( z )∣ w − z ∣ ) − p α ( df ( z )[ w − z ]∣ w − z ∣ ) ≤ p α ( f ( w ) − f ( z ) − df ( z )[ w − z ]∣ w − z ∣ ) < . It follows from the continuity of df ( z ) ∶ K d → ̂ E that there is C > such that p α ( f ( w ) − f ( z )∣ w − z ∣ ) < + p α ( df ( z )[ w − z ]∣ w − z ∣ ) ≤ + C ∣ w − z ∣∣ w − z ∣ = + C. Thus we have for all w ∈ K d with ∣ w − z ∣ < min ( ε /( + C ) , δ ) that p α ( f ( w ) − f ( z )) ≤ ( + C )∣ w − z ∣ < ε implying f ∈ C ( Ω , ̂ E ) . Since f ( Ω ) ⊂ E , we derive that f ∶ Ω → E is continuous. OLOMORPHIC 5 c) Let z ∈ Ω , v ∈ K d , α ∈ ̂ A and h ∈ K , h ≠ , such that z + hv ∈ Ω . We observethat p α ( f ( z + hv ) − f ( z ) h − df ( z )[ v ]) = ∣ v ∣ p α ( f ( z + hv ) − f ( z ) − df ( z )[ hv ]∣ hv ∣ ) which yields the Gâteaux-differentiability of f and Df ( z )[ v ] = df ( z )[ v ] because f is differentiable. Due to the linearity of df ( z ) we obtain for v = ( v , . . . , v d ) ∈ K d df ( z )[ v ] = d ∑ j = df ( z )[ e j ] v j = d ∑ j = Df ( z )[ e j ] v j = d ∑ j = ( ∂ e j K ) ̂ E f ( z ) v j . Finally, let ≤ j ≤ d and z ∈ π − z,j ( Ω ) ⊂ K . Clearly, v ↦ df ( π z,j ( z ))[ e j ] ⋅ v is alinear map from K to ̂ E and df ( π z,j ( ⋅ ))[ e j ] ⋅ v ∶ π − z,j ( Ω ) → ̂ E is continuous for every v ∈ C by the differentiability of f and the continuity of π z,j . We set ̃ z ∶ = π z,j ( z ) and get for w ∈ π − z,j ( Ω ) , w ≠ z , that p α ( ( f ○ π z,j )( w ) − ( f ○ π z,j )( z ) − df ( π z,j ( z ))[ e j ] ⋅ ( w − z )∣ w − z ∣ ) = p α ( f (̃ z + ( w − z ) e j ) − f (̃ z ) w − z − df (̃ z )[ e j ]) = p α ( f (̃ z + ( w − z ) e j ) − f (̃ z ) w − z − Df (̃ z )[ e j ]) . Letting w → z , we derive that f is separately differentiable and d ( f ○ π z,j )( z )[ v ] = df ( π z,j ( z ))[ e j ] ⋅ v , v ∈ K . d) We just have to observe that ( ̂ E ) ′ = E ′ by [24, 3.4.4 Corollary, p. 63] and getfor every e ′ ∈ E ′ d ( e ′ ○ f ) = e ′ ○ df, D ( e ′ ○ f ) = e ′ ○ Df, d ( e ′ ○ f ○ π z,j ) = e ′ ○ d ( f ○ π z,j ) for differentiable, Gâteaux-differentiable and separately differentiable f , respec-tively.e) Follows directly from Definition 3.1 c) and the fact that ̂ E is Hausdorff by [24,3.3.2 Theorem, p. 60]. (cid:3) We denote by φ ∶ C d → R d , φ ( Re z + i Im z , . . . , Re z d + i Im z d ) ∶ = ( Re z , Im z , . . . , Re z d , Im z d ) , the isometric isomorphism between C d and R d with respect to Euclidean norm onboth sides and remark the following.3.3. Remark.
Let E be an lcHs over C , Ω ⊂ C d open and f ∶ Ω → E holomorphic.Then f ○ φ − is real differentiable on φ ( Ω ) and d R ( f ○ φ − )( x )[ v ] = d C f ( φ − ( x ))[ φ − ( v )] , x ∈ φ ( Ω ) , v ∈ R d . (1)In particular, f ○ φ − ∈ C ( φ ( Ω ) , ̂ E ) . Proof. E is also an lcHs over R and equation (1) follows from ( f ○ φ − )( y ) − ( f ○ φ − )( x ) − d C f ( φ − ( x ))[ φ − ( x − y )]∣ x − y ∣ = f ( φ − ( y )) − f ( φ − ( x )) − d C f ( φ − ( x ))[ φ − ( x ) − φ − ( y )]∣ φ − ( x ) − φ − ( y )∣ for x, y ∈ φ ( Ω ) , x ≠ y , and the holomorphy of f on Ω . The R -linearity of d R ( f ○ φ − )( x ) for every x ∈ φ ( Ω ) and the continuity of d R ( f ○ φ − )( ⋅ )[ v ] ∶ φ ( Ω ) → ̂ E forevery v ∈ R d are a direct consequence of (1). K. KRUSE
Since f is continuous on Ω by Remark 3.2 b), the map f ○ φ − is continuous on φ ( Ω ) . Further, if e j is the j th unit vector in R d , we obtain ( ∂ e j R ) ̂ E ( f ○ φ − )( x ) = d R ( f ○ φ − )( x )[ e j ] = d C f ( φ − ( x ))[ φ − ( e j )] , x ∈ φ ( Ω ) . It follows that f ○ φ − is continuously partially (real) differentiable on φ ( Ω ) because d C f ( ⋅ )[ φ − ( e j )] is continuous for every ≤ j ≤ d . (cid:3) Our next goal is to derive Cauchy’s integral formula (for derivatives) for a holo-morphic function with values in a locally complete lcHs E . We use the notion of aPettis-integral to define integration of a vector-valued function.3.4. Definition (Pettis-integral) . Let Ω ⊂ R d , E an lcHs, ( Ω , L ( Ω ) , λ ) be themeasure space of Lebesgue measurable sets and L ( Ω , λ ) the space of K -valuedLebesgue-integrable (equivalence classes of) functions on Ω . A function f ∶ Ω → E is called weakly measurable if the function e ′ ○ f ∶ X → K , ( e ′ ○ f )( x ) ∶ = ⟨ e ′ , f ( x )⟩ ∶ = e ′ ( f ( x )) , is Lebesgue measurable for all e ′ ∈ E ′ . A weakly measurable functionis said to be weakly integrable if e ′ ○ f ∈ L ( Ω , λ ) . A function f ∶ Ω → E is called Pettis-integrable on Λ ∈ L ( Ω ) if it is weakly integrable on Λ and ∃ e Λ ( f ) ∈ E ∀ e ′ ∈ E ′ ∶ ⟨ e ′ , e Λ ( f )⟩ = ∫ Λ ⟨ e ′ , f ( x )⟩ d x. In this case e Λ ( f ) is unique due to E being Hausdorff and we define the Pettis-integral of f on Λ by ∫ Λ f ( x ) d x ∶ = e Λ ( f ) . A function γ ∶ [ a, b ] → C is called a C -curve (in C ) if γ can be extended to acontinuously differentiable function on an open set X ⊂ R with [ a, b ] ⊂ X . For afamily ( γ k ) ≤ k ≤ d of C -curves γ k ∶ [ a, b ] → C a function γ ∶ [ a, b ] d → C d , γ ( t , . . . , t d ) ∶ = ( γ ( t ) , . . . , γ d ( t d )) , is called a C -curve (in C d ) and we set l ( γ ) ∶ = ∫ [ a,b ] d d ∏ k = ∣ γ ′ k ( t k )∣ d t = d ∏ k = b ∫ a ∣ γ ′ k ( t k )∣ d t k which is the product of the length of the curves γ k . We say that γ is a C -curve in Ω ⊂ C d if there are open sets X k ⊂ R such that [ a, b ] ⊂ X k and γ k can be extendedto a continuously differentiable function ̃ γ k on X k for every ≤ k ≤ d and theso-defined extension ̃ γ ∶ = (̃ γ k ) k of γ on the open set X ∶ = ∏ ≤ k ≤ d X k ⊂ R d fulfils ̃ γ ( X ) ⊂ Ω .Let E be an lcHs over C , Ω ⊂ C d and γ ∶ [ a, b ] d → C d be a C -curve in Ω . Wedefine the (Pettis)-integral of a function f ∶ Ω → E along γ by ∫ γ f ( z ) d z ∶ = ∫ [ a,b ] d f ( γ ( t )) d ∏ k = γ ′ k ( t k ) d t if the Pettis-integral on the right-hand side exists. If the integral exists, we call f integrable along γ . Since γ is a C -curve in Ω , there is some open set X ⊂ R d suchthat [ a, b ] d ⊂ X and γ can be extended to a C -function ̃ γ on X with ̃ γ ( X ) ⊂ Ω . Ifthe extension of the factor of the integrand given by g ∶ X → E, g ( t ) ∶ = f (̃ γ ( t )) , is a weakly C function on X , we call f weakly γ - C . OLOMORPHIC 7
Proposition.
Let E be a locally complete lcHs over C , Ω ⊂ C d open, γ a C -curve in Ω and f ∶ Ω → E .a) If f is weakly γ - C , then f is integrable along γ .b) If f ○ φ − ∶ φ ( Ω ) → E is weakly C , then f is weakly γ - C in Ω .c) If f ∶ Ω → E is holomorphic, then f is weakly γ - C .Proof. a) As f is weakly γ - C , there is some open set X ⊂ R d such that [ a, b ] d ⊂ X and γ can be extended to a C -function ̃ γ on X with ̃ γ ( X ) ⊂ Ω so that f ○ ̃ γ isweakly C on X . We observe that ∣ I f ( e ′ )∣ ∶ = ∣ ∫ [ a,b ] d ⟨ e ′ , f ( γ ( t ))⟩ d ∏ k = γ ′ k ( t k ) d t ∣ ≤ l ( γ ) sup x ∈ f ( γ ([ a,b ] d )) ∣ e ′ ( x )∣ , e ′ ∈ E ′ . The closure of the absolutely convex hull acx f ( γ ([ a, b ] d )) = acx f (̃ γ ([ a, b ] d )) of f ( γ ([ a, b ] d )) is compact by [10, Proposition 2, p. 354] since f ○ ̃ γ is weakly C on X . Hence it follows that I f ∈ ( E ′ κ ) ′ and we deduce from the Mackey-Arens theoremthat there is e ( f ○ γ ) ∈ E such that ⟨ e ′ , e ( f ○ γ )⟩ = I f ( e ′ ) = ∫ [ a,b ] d ⟨ e ′ , f ( γ ( t ))⟩ d ∏ k = γ ′ k ( t k ) d t, e ′ ∈ E ′ , implying the integrability of f along γ .b) Indeed, writing e ′ ○ ( f ○ ̃ γ ) = ( e ′ ○ ( f ○ φ − )) ○ ( φ ○ ̃ γ ) , e ′ ∈ E ′ , for a C -extension ̃ γ of γ on X , we see that f ○ ̃ γ is weakly C on X by the scalarversion of the chain rule.c) We just have to notice that f is weakly holomorphic by Remark 3.2 d) implyingthat e ′ ○ ( f ○ φ − ) ∈ C ∞ ( φ ( Ω )) for all e ′ ∈ E ′ which proves the claim by part b). (cid:3) Next, we prove Fubini’s theorem which facilitates the computation of an integralalong a curve. We recall the following lemma whose proof is similar to the one ofProposition 3.5 a).3.6.
Lemma ([31, 4.7 Lemma, p. 14]) . Let E be a locally complete lcHs, Ω ⊂ R d open and f ∶ Ω → E . If f is weakly C , then f is Pettis-integrable (w.r.t. to theLebesgue measure) on every compact subset of K ⊂ Ω . Theorem (Fubini’s theorem) . Let E be a locally complete lcHs, Ω ⊂ R open, [ a, b ] × [ c, d ] ⊂ Ω and f ∶ Ω → E weakly C . Then f is Pettis-integrable on [ a, b ] × [ c, d ] and ∫ [ a,b ] × [ c,d ] f ( x , x ) d ( x , x ) = ∫ [ c,d ] ∫ [ a,b ] f ( x , x ) d x d x = ∫ [ a,b ] ∫ [ c,d ] f ( x , x ) d x d x . Proof.
The function f is Pettis-integrable on [ a, b ] × [ c, d ] by Lemma 3.6. Since Ω is open, there are ̃ a < a , b < ̃ b , ̃ c < c and d < ̃ d such that [̃ a, ̃ b ] × [̃ c, ̃ d ] ⊂ Ω . Further,we observe that F ∶ (̃ c, ̃ d ) → E, F ( x ) ∶ = ∫ [ a,b ] f ( x , x ) d x , is well-defined by Lemma 3.6 since f ( ⋅ , x ) is weakly C on (̃ a, ̃ b ) for every x ∈ (̃ c, ̃ d ) . We claim that F is weakly C on (̃ c, ̃ d ) . Indeed, we have ( e ′ ○ f )( ⋅ , x ) ∈L ([ a, b ]) and ( e ′ ○ f )( x , ⋅ ) ∈ C ((̃ c, ̃ d )) as well as ( e ′ ○ F )( x ) = ∫ [ a,b ] ( e ′ ○ f )( x , x ) d x (2) K. KRUSE for every x ∈ [ a, b ] , x ∈ (̃ c, ̃ d ) and e ′ ∈ E ′ . Furthermore, for every x ∈ (̃ c, ̃ d ) thereis ε > with B ε ( x ) = [ x − ε, x + ε ] ⊂ (̃ c, ̃ d ) and C e ′ ∶ = sup (∣ ∂ x ( e ′ ○ f )( x , ̃ x )∣ ∣ ( x , ̃ x ) ∈ [ a, b ] × B ε ( x )) < ∞ , e ′ ∈ E ′ , because e ′ ○ f ∈ C ( Ω ) for every e ′ ∈ E ′ . It follows from the scalar Leibniz rule fordifferentiation under the integral sign and the continuous dependency of a scalarintegral on a parameter (see [15, 5.6, 5.7 Satz, p. 147-148]) that e ′ ○ F ∈ C ( B ε ( x )) for every e ′ ∈ E ′ . As x ∈ (̃ c, ̃ d ) is arbitrary, we get that F is weakly C on (̃ c, ̃ d ) .Due to Lemma 3.6 again, we deduce that F is Pettis-integrable on [ c, d ] and thus ⟨ e ′ , ∫ [ c,d ] F ( x ) d x ⟩ = ∫ [ c,d ] ⟨ e ′ , F ( x )⟩ d x , e ′ ∈ E ′ . (3)Therefore we obtain for every e ′ ∈ E ′ that ⟨ e ′ , ∫ [ a,b ] × [ c,d ] f ( x , x ) d ( x , x )⟩ = ∫ [ a,b ] × [ c,d ] ⟨ e ′ , f ( x , x )⟩ d ( x , x ) = ∫ [ c,d ] ∫ [ a,b ] ⟨ e ′ , f ( x , x )⟩ d x d x = (2) ∫ [ c,d ] ⟨ e ′ , F ( x )⟩ d x = (3) ⟨ e ′ , ∫ [ c,d ] F ( x ) d x ⟩ = ⟨ e ′ , ∫ [ c,d ] ∫ [ a,b ] f ( x , x ) d x d x ⟩ where we used the scalar version of Fubini’s theorem in the second equation. TheHahn-Banach theorem yields the first equation from our claim and analogously weget the second equation ∫ [ a,b ] × [ c,d ] f ( x , x ) d ( x , x ) = ∫ [ a,b ] ∫ [ c,d ] f ( x , x ) d x d x . (cid:3) Fubini’s theorem for a continuous function f ∶ Ω ⊂ R → E can also be found in[11, Chap. 3, §4.1, Remark, p. INT III.43] by Bourbaki under the restriction that acx ( f ([ a, b ] × [ c, d ])) is compact in E . From the condition that f ∶ Ω → E is weakly C follows that f is continuous if E is sequentially complete or more general if E has metric ccp by [32, 6.4 Corollary, p. 19]. Thus in this case one can also applyBourbaki’s version of Fubini’s theorem.3.8. Remark.
Let E be a locally complete lcHs over C , Ω ⊂ C d open and γ a C -curve in Ω . If f ∶ Ω → E is weakly γ - C , then ∫ γ f ( z ) d z = ∫ [ a,b ] ⋯ ∫ [ a,b ] f ( γ ( t )) d ∏ k = γ ′ k ( t k ) d t d ⋯ d t by Fubini’s theorem.3.9. Proposition (chain rule) . Let E be an lcHs over C , Ω ⊂ C d open, γ ∶ [ a, b ] d → C d a C -curve in Ω and F ∶ Ω → E holomorphic. Then for every ≤ j ≤ d ( ∂ e j R ) ̂ E (( F ○ φ − ) ○ ( φ ○ γ ))( t ) = ( ∂ e j C ) ̂ E F ( γ ( t )) γ ′ j ( t j ) , t ∈ [ a, b ] d . (4) Proof.
Due to Remark 3.2 a)-c) and Remark 3.3 the map F ○ φ − ∶ φ ( Ω ) → E iscontinuous, the map D R ( F ○ φ − )( x ) ∶ R d → ̂ E is R -linear and the map D R ( F ○ φ − ) ∶ φ ( Ω ) × R d → ̂ E is continuous. The set φ ( Ω ) is open, thus for every x ∈ φ ( Ω ) there is R > such that B R ( x ) ⊂ φ ( Ω ) . Hence D R ( F ○ φ − ) is uniformly continuouson the compact set B R ( x ) × K for any compact set K ⊂ R d . Let ( ̂ E, ( p α ) α ∈ ̂ A ) OLOMORPHIC 9 denote the completion of E . It follows that for every α ∈ ̂ A and ε > there is δ > such that for all y ∈ B R ( x ) and v ∈ K with ∣ y − x ∣ = ∣( y, v ) − ( x, v )∣ < δ we have sup v ∈ K p α ( D R ( F ○ φ − )( y )[ v ] − D R ( F ○ φ − )( x )[ v ]) < ε implying that D R ( F ○ φ − ) ∶ φ ( Ω ) → L c ( R d , ̂ E ) is continuous. Since γ is a C -curve in Ω , there are an open set X ⊂ R d with [ a, b ] d ⊂ X and a continuously partially differentiable extension ̃ γ of γ on X such that φ ○ ̃ γ ∶ X → R d is continuous, D R ( φ ○ ̃ γ ) ∶ X → L ( R d , R d ) and by direct computation D R ( φ ○ ̃ γ )( x )[ v ] = d ∑ k = (( Re ̃ γ k ) ′ ( x k ) e k − + ( Im ̃ γ k ) ′ ( x k ) e k ) v k , x ∈ X, v ∈ R d . For x, y ∈ X we set u k ∶ = ( Re ̃ γ k ) ′ ( y k ) − ( Re ̃ γ k ) ′ ( x k ) and w k ∶ = ( Im ̃ γ k ) ′ ( y k ) − ( Im ̃ γ k ) ′ ( x k ) and observe for v ∈ R d that ∣ D R ( φ ○ ̃ γ )( y )[ v ] − D R ( φ ○ ̃ γ )( x )[ v ]∣ = ∣ d ∑ k = ( u k e k − + w k e k ) v k ∣ = ( d ∑ k = ( u k + w k ) v k ) / ≤ d ∑ k = ( u k + w k ) / ∣ v k ∣ ≤ ∣ d ∑ k = ( u k + w k ) / e k ∣ ⋅ ∣ v ∣ = d ∑ k = ∣̃ γ ′ k ( y k ) − ̃ γ ′ k ( x k )∣ ⋅ ∣ v ∣ where the second inequality follows from the Cauchy-Schwarz inequality. Thisimplies that D R ( φ ○ ̃ γ ) ∶ X → L b ( R d , R d ) is continuous because γ is continuouslypartially differentiable. Summarising, this means that F ○ φ − and φ ○ ̃ γ are ofclass C k in the notion of [28, 1.0.0 Definition, p. 59]. From (the proof of) [28, 1.3.4Corollary, p. 80] follows that D R ( F ○ φ − )( φ ○ ̃ γ ( ⋅ ))[ D R ( φ ○ ̃ γ )( ⋅ )] ∶ X → L c ( R d , ̂ E ) is continuous and thus D R (( F ○ φ − ) ○ ( φ ○ ̃ γ ))( x )[ v ] = D R ( F ○ φ − )( φ ○ ̃ γ ( x ))[ D R ( φ ○ ̃ γ )( x )[ v ]] , x ∈ X, v ∈ R d , by the chain rule [28, 1.3.0 Theorem, p. 77]. In combination with Remark 3.2 c)we obtain for every x ∈ X and v ∈ R d that d ∑ k = ( ∂ e k R ) ̂ E (( F ○ φ − ) ○ ( φ ○ ̃ γ ))( x ) v k = D R (( F ○ φ − ) ○ ( φ ○ ̃ γ ))( x )[ v ] = (1) D C F (( φ − ○ φ ○ ̃ γ )( x ))[ φ − ( D R ( φ ○ ̃ γ )( x )[ v ])] = D C F (̃ γ ( x ))[ d ∑ k = ̃ γ ′ k ( x k ) e k v k ] = d ∑ k = ( ∂ e k C ) ̂ E F (̃ γ ( x ))̃ γ ′ k ( x k ) v k and thus with v = e j , ≤ j ≤ d , ( ∂ e j R ) ̂ E (( F ○ φ − ) ○ ( φ ○ ̃ γ ))( x ) = ( ∂ e j C ) ̂ E F (̃ γ ( x ))̃ γ ′ j ( x j ) connoting (4) for x ∈ [ a, b ] d . (cid:3) Theorem (fundamental theorem of calculus) . Let E be a locally completelcHs over C , Ω ⊂ C open, γ ∶ [ a, b ] → C a C -curve in Ω , f ∶ Ω → E weakly γ - C andlet there be a holomorphic function F ∶ Ω → E such that F ′ = f . Then ∫ γ f ( z ) d z = F ( γ ( b )) − F ( γ ( a )) . (5) Proof.
The left-hand side of (5) is defined by Proposition 3.5 a). Due to the chainrule Proposition 3.9 and Remark 3.2 e) we have ∫ γ f ( z ) d z = ∫ [ a,b ] f ( γ ( t )) γ ′ ( t ) d t = ∫ [ a,b ] f ( γ ( t ))´¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¶ = F ′ ( γ ( t )) γ ′ ( t ) d t = (4) ∫ [ a,b ] (( F ○ φ − ) ○ ( φ ○ γ )) ′ ( t ) d t. Looking at the last integral, we observe that ⟨ e ′ , ∫ [ a,b ] (( F ○ φ − ) ○ ( φ ○ γ )) ′ ( t ) d t ⟩ = b ∫ a ( e ′ ○ ( F ○ φ − ) ○ ( φ ○ γ )) ′ ( t ) d t = ( e ′ ○ ( F ○ φ − ) ○ ( φ ○ γ ))( b ) − ( e ′ ○ ( F ○ φ − ) ○ ( φ ○ γ ))( a ) = ⟨ e ′ , ( F ○ γ )( b ) − ( F ○ γ )( a )⟩ , e ′ ∈ E ′ , holds by the scalar fundamental theorem of calculus (applied to the real and theimaginary part of the integrand) where the second integral is a Riemann-integral.Finally, we deduce from the Hahn-Banach theorem that ∫ γ f ( z ) d z = F ( γ ( b )) − F ( γ ( a )) . (cid:3) Lemma (Leibniz’ rule) . Let E be a locally complete lcHs over C , V, U ⊂ C d open and γ ∶ [ a, b ] d → C d a C -curve in V .a) Let T be a set and f, f n ∶ V × T → E such that f ( ⋅ , t ) , f n ( ⋅ , t ) ∶ V → E areweakly γ - C for every t ∈ T , n ∈ N and f n → f uniformly on γ ([ a, b ] d ) × T .Then lim n →∞ ∫ γ f n ( z, t ) d z = ∫ γ f ( z, t ) d z holds uniformly on T .b) Let f ∶ V × U → E be such that f ( ⋅ , λ ) ∶ V → E is weakly γ - C for every λ ∈ U , f ( z, ⋅ ) ∶ U → E is holomorphic for every z ∈ V with ( ∂ λ j ) E f ∶ V × U → E beingcontinuous and ( ∂ λ j ) E f ( ⋅ , λ ) ∶ V → E weakly γ - C for every λ ∈ U and some ≤ j ≤ d . Then G ∶ U → E, G ( λ ) ∶ = ∫ γ f ( z, λ ) d z, is well-defined, complex differentiable with respect to λ j and ( ∂ e j C ) E G ( λ ) = ∫ γ ( ∂ λ j ) E f ( z, λ ) d z ∈ E, λ ∈ U. Proof.
First, we remark that the integrals appearing in a) and b) are well-definedelements of E by Proposition 3.5 a) and the weakly γ - C condition.a) Let α ∈ A . Then we have sup t ∈ T p α ( ∫ γ f n ( z, t ) d z − ∫ γ f ( z, t ) d z ) ≤ l ( γ ) sup ( z,t ) ∈ γ ([ a,b ] d ) × T p α ( f n ( z, t ) − f ( z, t )) → , n → ∞ , OLOMORPHIC 11 since f n → f uniformly on γ ([ a, b ] d ) × T .b) Let λ ∈ U . Then there is R > such that B R ( λ ) ⊂ U as U is open. Let ( h n ) bea null sequence in C ∖ { } with ∣ h n ∣ < R / for all n ∈ N which implies that the linesegment Γ n from λ j to λ j + h n is a C -curve in π − λ,j ( B R ( λ )) that we parametriseby [ , ] . Applying Theorem 3.10 to the holomorphic function (in one variable) f ( z, ⋅ ) ○ π λ,j ∶ π − λ,j ( B R ( λ )) → E for z ∈ V , we get f ( z, λ + h n e j ) − f ( z, λ ) = f ( z, π λ,j ( λ j + h n )) − f ( z, π λ,j ( λ j )) = ∫ Γ n ( ∂ ζ j ) E f ( z, π λ,j ( ζ j )) d ζ j and therefore ∣ f n ( z, λ )∣ ∶ = ∣ f ( z, λ + h n e j ) − f ( z, λ ) h n − ( ∂ λ j ) E f ( z, λ )∣ = ∣ h n ∫ Γ n ( ∂ ζ j ) E f ( z, π λ,j ( ζ j )) − ( ∂ λ j ) E f ( z, λ ) d ζ j ∣ ≤ ∣ h n ∣ l ( Γ n ) sup ζ j ∈ Γ n ([ , ]) ∣( ∂ ζ j ) E f ( z, π λ,j ( ζ j )) − ( ∂ λ j ) E f ( z, λ )∣ . Hence we obtain sup z ∈ γ ([ a,b ] d ) ∣ f n ( z, λ )∣ ≤ sup z ∈ γ ([ a,b ] d ) sup ζ j ∈ Γ n ([ , ]) ∣( ∂ ζ j ) E f ( z, π λ,j ( ζ j )) − ( ∂ λ j ) E f ( z, λ )∣ → , n → ∞ , since ( ∂ λ j ) E f is uniformly continuous on the compact set γ ([ a, b ] d ) × B R ( λ ) meaning f n → uniformly on γ ([ a, b ] d ) × { λ } . From part a) we conclude ∫ γ f n ( z, λ ) d z → and thus ∫ γ ( ∂ λ j ) E f ( z, λ ) d z = lim n →∞ ∫ γ f ( z, λ + h n e j ) − f ( z, λ ) h n d z = lim n →∞ G ( z, λ + h n e j ) − G ( z, λ ) h n = ( ∂ e j C ) E G ( λ ) . (cid:3) Now, we want to define complex partial derivatives of higher order for an E -valued function f . Let E be an lcHs over C and Ω ⊂ C d open. A function f ∶ Ω → E is called complex partially differentiable on Ω and we write f ∈ D C ( Ω , E ) if ∂ e j C f ( z ) ∶ = ( ∂ e j C ) E f ( z ) ∈ E for every z ∈ Ω and ≤ j ≤ d (see Definition 3.1 c)). For k ∈ N , k ≥ , a function f is said to be k -times complex partially differentiable andwe write f ∈ D k C ( Ω , E ) if f ∈ D C ( Ω , E ) and all its first complex partial derivativesare in D k − C ( Ω , E ) . A function f is called infinitely complex partially differentiableand we write f ∈ D ∞ C ( Ω , E ) if f ∈ D k C ( Ω , E ) for every k ∈ N .Let f ∈ D k C ( Ω , E ) . For β = ( β , . . . , β d ) ∈ N d with ∣ β ∣ ∶ = ∑ dj = β j ≤ k we set ∂ β j C f ∶ = ( ∂ β j C ) E f ∶ = f , if β j = , and ∂ β j C f ∶ = ( ∂ β j C ) E f ∶ = ( ∂ e j C ) ⋯ ( ∂ e j C )´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ β j -times f, if β j ≠ , as well as ∂ β C f ∶ = ( ∂ β C ) E f ∶ = ( ∂ β C ) ⋯ ( ∂ β d C ) f. A holomorphic function f ∶ Ω → E can be considered as a function from Ω to ̂ E which gives us f ∈ D C ( Ω , ̂ E ) (see Remark 3.2 c)). Our goal is to show that weactually have f ∈ D ∞ C ( Ω , E ) if E is locally complete via proving Cauchy’s integralformula for holomorphic functions. For this purpose we recall the definition of apolydisc, its distinguished boundary and define integration along the distinguishedboundary. For w = ( w , . . . , w d ) ∈ C d and R = ( R , . . . , R d ) ∈ ( , ∞ ] d we definethe polydisc D R ( w ) ∶ = ∏ dk = B R k ( w k ) and its distinguished boundary ∂ D R ( w ) ∶ = ∏ dk = ∂ B R k ( w k ) . For R, ρ ∈ ( , ∞ ] d we write ρ < R if ρ k < R k for all ≤ k ≤ d . For afunction f ∶ Ω → E on a set Ω ⊂ C d with D ρ ( w ) ⊂ Ω for some w ∈ C d and ρ ∈ ( , ∞ ) d we set ∫ ∂ D ρ ( w ) f ( z ) d z ∶ = ∫ γ f ( z ) d z if the integral on the right-hand side exists where γ is the C -curve in Ω given bythe restriction γ ∶ = ̃ γ ∣[ , π ] d of the map ̃ γ ∶ R d → C d defined by ̃ γ k ∶ R → C , ̃ γ k ( t ) ∶ = w k + ρ k e it , for ≤ k ≤ d . Further, we need the usual notation β ! ∶ = d ∏ j = ( β j ! ) and ( z − ζ ) β ∶ = d ∏ j = ( z j − ζ j ) β j for β ∈ N d and z, ζ ∈ C d .3.12. Theorem (Cauchy’s integral formula) . Let E be a locally complete lcHs over C , Ω ⊂ C d open, w ∈ Ω , R ∈ ( , ∞ ] d with D R ( w ) ⊂ Ω and f ∶ Ω → E be holomorphic.Then ( ∂ β C ) f ( ζ ) = β ! ( πi ) d ∫ ∂ D ρ ( w ) f ( z )( z − ζ ) β + ( ,..., ) d z ∈ E, ζ ∈ D ρ ( w ) , β ∈ N d , (6) for all ρ ∈ ( , ∞ ) d with ρ < R .Proof. Let ̃ γ and γ be defined as above for ∂ D ρ ( w ) . First, we consider the case β = . We set g ζ ∶ Ω ∖ { ζ } → E, g ζ ( z ) ∶ = f ( z )( z − ζ ) ( ,..., ) , for ζ ∈ D ρ ( w ) and observe that g ζ ○ ̃ γ is weakly C on R d since f is holomorphic on Ω and ̃ γ ∈ C ( R d , C d ) . Thus g ζ is weakly γ - C and integrable along γ by Proposition3.5 a). Since f is weakly holomorphic and g ζ integrable along γ , we get by thescalar version of Cauchy’s integral formula that ( e ′ ○ f )( ζ ) = ( πi ) d ∫ ∂ D ρ ( w ) ( e ′ ○ f )( z )( z − ζ ) ( ,..., ) d z = ⟨ e ′ , ( πi ) d ∫ ∂ D ρ ( w ) f ( z )( z − ζ ) ( ,..., ) d z ⟩ , e ′ ∈ E ′ , implying f ( ζ ) = ( πi ) d ∫ ∂ D ρ ( w ) f ( z )( z − ζ ) ( ,..., ) d z by the Hahn-Banach theorem which proves (6) for β = .Let n ∈ N and (6) be fullfilled for every β ∈ N d with ∣ β ∣ = n . Let β ∈ N d with ∣ β ∣ = n + . Then there is j ∈ N , ≤ j ≤ d , and ̃ β ∈ N d with ∣̃ β ∣ = n such that β = ̃ β + e j . OLOMORPHIC 13
Let ζ ∈ D ρ ( w ) . Then there is < r < ρ such that ζ ∈ D r ( w ) . We define the open set V ∶ = D R ( w ) ∖ D r ( w ) and the function F ̃ β ∶ V × D r ( w ) → E, F ̃ β ( z, λ ) ∶ = f ( z )( z − λ ) ̃ β + ( ,..., ) . Further, we compute for λ ∈ D r ( w ) ∂ λ j F ̃ β ( z, λ ) = (̃ β j + ) f ( z )( z − λ ) ̃ β + e j + ( ,..., ) = β j f ( z )( z − λ ) β + ( ,..., ) ∈ E, z ∈ V. We see that γ is a C -curve in V and F ( ⋅ , λ ) ○ ̃ γ and ∂ λ j F ̃ β ( ⋅ , λ ) ○ ̃ γ are weakly C on R d for every λ ∈ D r ( w ) since f is holomorphic on Ω . Hence F ̃ β ( ⋅ , λ ) and ∂ λ j F ̃ β ( ⋅ , λ ) are weakly γ - C for every λ ∈ D r ( w ) . In addition, ∂ λ j F ̃ β is continuous on V × D r ( w ) by Remark 3.2 b), F ̃ β ( z, ⋅ ) is holomorphic on D r ( w ) for every z ∈ V and thus wecan apply Leibniz’ rule Lemma 3.11 b) yielding ∂ e j C ( ∂ ̃ β C f )( λ ) = ̃ β ! ( πi ) d ∫ γ ∂ λ j F ̃ β ( z, λ ) d z = β ! ( πi ) d ∫ γ f ( z )( z − λ ) β + ( ,..., ) d z ∈ E for every λ ∈ D r ( w ) , in particular for λ = ζ , where we used the induction hypothesisin the first equation. It remains to be shown that ∂ e j C ( ∂ ̃ β C f )( λ ) = ∂ β C f ( λ ) for every λ ∈ D r ( w ) , i.e. that the order of the partial derivatives does not matter. For ̃ β = this is clear. If ∣̃ β ∣ = , then our preceding considerations imply that ∂ e j C ∂ e k C f ( λ ) = ( πi ) d ∫ ∂ D ρ ( w ) f ( z )( z − λ ) e j + e k + ( ,..., ) d z = ∂ e k C ∂ e j C f ( λ ) for all ≤ j, k ≤ d . This yields that ∂ e j C ( ∂ ̃ β C f )( λ ) = ∂ β C f ( λ ) for every λ ∈ D r ( w ) (cid:3) Cauchy’s integral formula for derivatives is usually derived by using the Riemann-integral instead of the Pettis-integral and can be found for holomorphic functionsin one variable in [21, Théorème 1, p. 37-38], in several variables in [7, Corollary3.7, p. 85] and infinitely many variables in [12, Proposition 2.4, p. 55] as well. TheRiemann-integrals are elements of E under the condition that E has ccp in [21]or more general if E is sequentially complete in [7] and [12] by [7, Lemma 1.1, p.79]. In general, they are only elements of the completion ̂ E . From our approachusing Pettis-integrals we guarantee that they belong to E even if E is only locallycomplete.3.13. Corollary. If E is a locally complete lcHs over C , Ω ⊂ C d open and f ∶ Ω → E holomorphic, then ( ∂ β C ) f does not depend on the order of the partial derivativesinvolved and f ∈ D ∞ C ( Ω , E ) .Proof. The independence of the order follows from the proof of Cauchy’s integralformula Theorem 3.12. Combining this formula with the commutativity of thecomplex partial derivatives, we conclude f ∈ D ∞ C ( Ω , E ) . (cid:3) For an lcHs E over C , an open set Ω ⊂ C d and a function f ∶ Ω → E , we write f ∈ C k R ( Ω , E ) if f ○ φ − ∈ C k ( φ ( Ω ) , E ) for k ∈ N , ∞ . We define the space O ( Ω , E ) ∶ = { f ∈ C R ( Ω , E ) ∣ ∀ β ∈ N d , z ∈ Ω ∶ ( ∂ β C ) E f ( z ) ∈ E } which we equip with the system of seminorms given by ∣ f ∣ K,α ∶ = sup z ∈ K p α ( f ( z )) , f ∈ O ( Ω , E ) , for K ⊂ Ω compact and α ∈ A . If E = C , we just write O ( Ω ) ∶ = O ( Ω , C ) . Due to Cauchy’s integral formula and Remark 3.3 in combination with Remark3.2 c)+e), we already know that every holomorphic function f ∶ Ω → E is an ele-ment of O ( Ω , E ) and we prove in the following that every element of O ( Ω , E ) isa holomorphic function on Ω as well if E is locally complete. The space O ( Ω ) coincides with the space of all C -valued holomorphic functions on Ω in the sense of[25, Definition 1.7.1, p. 47] by [25, Theorem 1.7.6, p. 48-49] and is a Fréchet spaceby [25, Example 1.10.7 (a), p. 66]. As a start in proving that O ( Ω , E ) is the spaceof all holomorphic functions from Ω to a locally complete space E , we show thatthe elements of O ( Ω , E ) fulfil the Cauchy inequality.3.14. Corollary (Cauchy inequality) . Let E be a locally complete lcHs over C and Ω ⊂ C d open.a) If w ∈ Ω , R ∈ ( , ∞ ] d with D R ( w ) ⊂ Ω and f ∈ O ( Ω , E ) , then p α ( ∂ β C f ( ζ )) ≤ β ! ρ β max z ∈ ∂ D ρ ( w ) p α ( f ( z )) , ζ ∈ D ρ ( w ) , β ∈ N d , (7) for every ρ ∈ ( , ∞ ) d with ρ < R and α ∈ A .b) For every compact set K ⊂ Ω there is a compact set K ′ ⊂ Ω such thatfor every β ∈ N d there is C K,β > such that for every α ∈ A and every f ∈ O ( Ω , E ) holds sup z ∈ K p α ( ∂ β C f ( z )) ≤ C K,β max z ∈ K ′ p α ( f ( z )) . (8) Proof. a) For α ∈ A we set B α ∶ = { x ∈ E ∣ p α ( x ) < } , its polar B ○ α ∶ = { e ′ ∈ E ′ ∣ ∀ x ∈ B α ∶ ∣ e ′ ( x )∣ ≤ } and denote by γ the C -curve on [ , π ] d correspondingto ∂ D ρ ( w ) . It follows from the scalar version of Cauchy’s integral formula (see [25,Theorem 1.7.6, p. 48-49]) that for all ζ ∈ D ρ ( w ) and β ∈ N d we have p α (( ∂ β C ) E f ( ζ )) = sup e ′ ∈ B ○ α ∣( ∂ β C ) C ( e ′ ○ f )( ζ )∣ = β ! ( π ) d sup e ′ ∈ B ○ α ∣ ∫ ∂ D ρ ( w ) e ′ ( f ( z ))( z − ζ ) β + ( ,..., ) d z ∣ = β ! ( π ) d sup e ′ ∈ B ○ α ∣ ∫ ∂ D ρ ( w ) e ′ ( f ( z ))( z − ζ ) β + ( ,..., ) d z ∣ ≤ β ! ( π ) d l ( γ )´¸¶ = ( π ) d ρ ( ,..., ) sup e ′ ∈ B ○ α sup z ∈ ∂ D ρ ( w ) ∣ e ′ ( f ( z ))∣ ρ β + ( ,..., ) = β ! ρ β sup z ∈ ∂ D ρ ( w ) p α ( f ( z )) where we used [33, Proposition 22.14, p. 256] in the first and last equation to getfrom p α to sup e ′ ∈ B ○ α and back. Our statement follows from the continuity of f on Ω by Remark 3.2 b) and the compactness of the distinguished boundary.b) Is a direct consequence of a) since every compact set K ⊂ Ω can be coveredby a finite number n of open, bounded polydiscs D ρ j ( w j ) with D ρ j ( w j ) ⊂ Ω for ≤ j ≤ n . (cid:3) For sequentially complete E Cauchy’s inequality can also be found in [12, Propo-sition 2.5, p. 57] and as a direct consequence we obtain:3.15.
Remark (Weierstrass) . Let E be a locally complete lcHs over C and Ω ⊂ C d open. Then the system of seminorms generated by ∣ f ∣ K,m,α ∶ = sup z ∈ Kβ ∈ N d , ∣ β ∣ ≤ m p α (( ∂ β C ) E f ( z )) , f ∈ O ( Ω , E ) , OLOMORPHIC 15 for K ⊂ Ω compact, m ∈ N and α ∈ A induces the same topology on O ( Ω , E ) asthe system (∣ f ∣ K,α ) by (8).This remark implies [7, Proposition 3.1, p. 85] if E is sequentially complete.We observe the following useful relation between real and complex first partialderivatives.3.16. Proposition. If E is an lcHs over C , Ω ⊂ C d open and f ∈ O ( Ω , E ) , then forevery ≤ j ≤ d and x ∈ φ ( Ω ) ∂ e j R ( f ○ φ − )( x ) = i∂ e j C f ( φ − ( x )) and ∂ e j − R ( f ○ φ − )( x ) = ∂ e j C f ( φ − ( x )) . Proof. f ∈ C R ( Ω , E ) and for x = ( x , . . . , x d ) ∈ φ ( Ω ) we get ∂ e j R ( f ○ φ − )( x ) = lim h → h ∈ R ,h ≠ f ( . . . , x j − + ix j + ih, . . . ) − f ( . . . , x j − + ix j , . . . ) h = i lim h → h ∈ R ,h ≠ f ( . . . , x j − + ix j + ih, . . . ) − f ( . . . , x j − + ix j , . . . ) ih = i∂ e j C f ( φ − ( x )) as well as ∂ e j − R ( f ○ φ − )( x ) = lim h → h ∈ R ,h ≠ f ( . . . , x j + ix j + + h, . . . ) − f ( . . . , x j + ix j + , . . . ) h = ∂ e j C f ( φ − ( x )) . (cid:3) Proposition.
Let E be a locally complete lcHs over C and Ω ⊂ C d open. Thenthe map S ∶ O ( Ω ) εE → O ( Ω , E ) , u z→ [ z ↦ u ( δ z )] , is a (topological) isomorphism where δ z is the point evaluation functional at z .Proof. Let u ∈ O ( Ω ) εE . Due to [30, 4.12 Proposition, p. 22] and the barrellednessof the Fréchet space O ( Ω ) we have ( ∂ β C ) E S ( u )( z ) = u ( δ z ○ ∂ β C ) ∈ E for all β ∈ N d and z ∈ Ω where one has to replace ∂ β R by ∂ β C and the space CW k ( Ω ) by O ( Ω ) inthe proof of [30, 4.12 Proposition, p. 22]. Furthermore, S ( u ) ∈ C R ( Ω , E ) by [30,4.12 Proposition, p. 22] with k = , Remark 3.15 and Proposition 3.16 which impliesthat S ( u ) ∈ O ( Ω , E ) .Let f ∈ O ( Ω , E ) and K ⊂ Ω be compact. It is easily checked that e ′ ○ f ∈ O ( Ω ) for every e ′ ∈ E ′ . Since f ○ φ − is weakly C on the open set φ ( Ω ) ⊂ R d , it followsfrom [10, Proposition 2, p. 354] that K ∶ = acx ( f ( K )) is a absolutely convex andcompact. The inclusion N K ( f ) ∶ = f ( K ) ⊂ K implies that [30, 3.16 Condition d), p.12] is fulfilled yielding that S is a (topological) isomorphism by [30, 3.17 Theorem,p. 12] in combination with [30, 3.13 Lemma b), p. 10]. (cid:3) Once we have the equivalent conditions for holomorphy from our main Theo-rem 3.20, namely the euqivalence ‘ a ) ⇔ d ) ’, the preceding proposition is just aconsequence of [9, Theorem 9, p. 232].3.18. Theorem.
Let E be a locally complete lcHs over C , z ∈ C d and R ∈ ( , ∞ ] d .Then the tensor product O ( D R ( z )) ⊗ E is sequentially dense in O ( D R ( z ) , E ) and f = ∑ β ∈ N d ( ∂ β C ) E f ( z ) β ! ( ⋅ − z ) β for all f ∈ O ( D R ( z ) , E ) where the series converges in O ( D R ( z ) , E ) . Proof.
The monomials ( ⋅ − z ) β , β ∈ N d , form an equicontinuous Schauder basis withassociated coefficient functionals β ! ( δ z ○ ∂ β C ) of the barrelled space O ( D R ( z )) by[25, Theorem 1.7.6, p. 48-49]. Thus our statement follows from Proposition 3.17and [31, 3.6 Corollary, p. 7]. (cid:3) In the one variable case the theorem above is given in [31, 3.6 Corollary c), p. 7]combined with [31, 4.5 Theorem, p. 13] as well.3.19.
Corollary.
Let E be a locally complete lcHs over C and Ω ⊂ C d open. Thenthe following statements are equivalent for a function f ∶ Ω → E .a) f is holomorphic on Ω .b) f ∈ O ( Ω , E ) .Proof. We only need to prove the implication ‘ b ) ⇒ a ) ’. We claim that for every z ∈ Ω holds df ( z )[ v ] = d ∑ j = ∂ e j C f ( z ) v j , v ∈ C d . (9)Observe that the right-hand side is already linear in v . Let α ∈ A and z ∈ Ω . Thenthere is R ∈ ( , ∞ ] d such that D R ( z ) ⊂ Ω . We fix ρ ∈ ( , ∞ ) d with ρ < R and derivefrom Theorem 3.18 a) that g ( w, z ) ∶ = f ( w ) − f ( z ) − d ∑ j = ∂ e j C f ( z )( w j − z j ) = ∑ ∣ β ∣ > ∂ β C f ( z ) β ! ( w − z ) β = d ∑ j = ( w j − z j ) ∑ ∣ β ∣ ≥ ∂ β + e j C f ( z )( β + e j ) ! ( w − z ) β for every w ∈ D ρ ( z ) .Let < ε ≤ and set r ∶ = min ≤ j ≤ d ρ j . We observe that B ε ⋅ r / ( z ) is a subset of D ρ ( z ) . Applying Cauchy’s inequality (7), we obtain for w ∈ B ε ⋅ r / ( z ) p α ( ∂ β + e j C f ( z )( β + e j ) ! ( w − z ) β ) ≤ ∏ dk = ∣ w k − z k ∣ β k ρ β + e j max ζ ∈ ∂ D ρ ( z ) p α ( f ( ζ )) ≤ r max ζ ∈ ∂ D ρ ( z ) p α ( f ( ζ )) d ∏ k = ( ∣ w − z ∣ r ) β k ≤ r max ζ ∈ ∂ D ρ ( z ) p α ( f ( ζ )) d ∏ k = ε β k β k ≤ ε d ∣ β ∣ r max ζ ∈ ∂ D ρ ( z ) p α ( f ( ζ )) . Hence we conclude for every w ∈ B ε ⋅ r / ( z ) p α ( g ( w, z )∣ w − z ∣ ) ≤ d ⋅ ε d r max ζ ∈ ∂ D ρ ( z ) p α ( f ( ζ )) ∑ ∣ β ∣ ≥ ∣ β ∣ ≤ d ⋅ d ⋅ ε d r max ζ ∈ ∂ D ρ ( z ) p α ( f ( ζ )) where the last estimate follows from [25, Corollary 1.2.14 (a), p. 12-13]. Letting ε → , proves (9).Fix v ∈ C d and let z, w ∈ Ω . The estimate p α ( df ( w )[ v ] − df ( z )[ v ]) ≤ d ∑ j = p α ( ∂ e j C f ( w ) − ∂ e j C f ( z ))∣ v j ∣ implies that df ( ⋅ )[ v ] is continuous on Ω since f ∈ C R ( Ω ) and by Proposition 3.16.Therefore f is holomorphic on Ω . (cid:3) We briefly recall the following definitions which enable us to phrase our maintheorem concerning holomorphic functions in several variables. Let E be an lcHs OLOMORPHIC 17 over C . For an open set Ω ⊂ R d and ≤ j ≤ d we define the Cauchy-Riemannoperator by ∂ j f ( x ) ∶ = ( ∂ j ) E f ( x ) ∶ = ( ∂ e j − R + i∂ e j R ) f ( x ) , f ∈ C ( Ω , E ) , x ∈ Ω . A function f ∶ Ω → E from a topological space Ω to E is called locally bounded on asubset Λ ⊂ Ω if for every z ∈ Λ there is a neighbourhood U ⊂ Ω of z such that f isbounded on U . A subspace G ⊂ E ′ is said to be separating if for every x, y ∈ E thereis e ′ ∈ G such that e ′ ( x ) ≠ e ′ ( y ) . A subspace G ⊂ E ′ is said to determine boundedness if every σ ( E, G ) -bounded subset of E is already bounded where σ ( E, G ) denotesthe weak topology w.r.t. the dual pair ⟨ E, G ⟩ . If G determines boundedness, then G is separating. For instance, G ∶ = E ′ determines boundedness by Mackey’s theoremand further examples may be found in [1, Remark 1.4, p. 781-782] and [9, Remark11, p. 233].3.20. Theorem.
Let E be a locally complete lcHs over C and Ω ⊂ C d be open. Thenthe following statements are equivalent for a function f ∶ Ω → E .a) f is holomorphic (Gâteaux-, separately holomorphic) on Ω .b) f ∈ C ( Ω , E ) and ∂ e j C f ( z ) exists in ̂ E for every z ∈ Ω and ≤ j ≤ d .c) ∂ e j C f ( z ) exists in ̂ E for every z ∈ Ω and ≤ j ≤ d .d) f ∈ C ∞ R ( Ω , E ) and ∂ j ( f ○ φ − ) = for all ≤ j ≤ d .e) There is a subspace G ⊂ E ′ which determines boundedness such that e ′ ○ f is holomorphic (Gâteaux-, separately holomorphic) on Ω for every e ′ ∈ G .f ) f is locally bounded outside some compact set K ⊂ Ω and there is a sepa-rating subspace G ⊂ E ′ such that e ′ ○ f is holomorphic (Gâteaux-, separatelyholomorphic) on Ω for every e ′ ∈ G .g) For every z ∈ Ω there are R ∈ ( , ∞ ] d and ( a β ) β ∈ N d ⊂ E such that f = ∑ β ∈ N d a β ( ⋅ − z ) β on D R ( z ) . If one of the equivalent conditions above is fulfilled, then df ( z )[ v ] = Df ( z )[ v ] = d ∑ j = ∂ e j C f ( z ) v j ∈ E, z ∈ Ω , v ∈ C d . (10) Proof.
We write x )( i ) if we consider x ) for holomorphic functions, x )( ii ) if weconsider x ) for Gâteaux-holomorphic functions and x )( iii ) if we consider x ) forseparately holomorphic functions in the cases x ∈ { a, e, f } . First, we remark thatthe endorsement (10) follows from Cauchy’s integral formula and Remark 3.2 c)+e).‘ a )( i ) ⇔ e )( i ) ’: The implication ‘ ⇒ ’ is clear with G ∶ = E ′ . Let us turn to ‘ ⇐ ’.We claim that O ( Ω , E ) coincides with the space of functions f ∶ Ω → E such that e ′ ○ f ∈ O ( Ω ) for each e ′ ∈ G which then yields the desired equivalence by Corollary3.19. O ( Ω ) is a closed subspace of C ∞ ( φ ( Ω )) via the map f ↦ f ○ φ − (see e.g. [18,p. 691]). Thus we can apply the weak-strong principle [9, Corollary 10 (a), p. 233]in combination with [9, Definition 3, p. 229-230] and Proposition 3.17 proving ourclaim.‘ a )( i ) ⇒ g ) ’: Follows from Corollary 3.19 and Theorem 3.18 since for every f ∈O ( Ω , E ) there is z ∈ Ω and R ∈ ( , ∞ ] d such that f ∣ D R ( z ) ∈ O ( D R ( z ) , E ) .‘ g ) ⇒ e )( i ) ’: Let z ∈ Ω , R ∈ ( , ∞ ] d and ( a β ) β ∈ N d ⊂ E be such that f ( w ) = ∑ β ∈ N d a β ( w − z ) β , w ∈ D R ( z ) . Then we have for every e ′ ∈ G ∶ = E ′ that ( e ′ ○ f )( w ) = ∑ β ∈ N d e ′ ( a β )( w − z ) β , w ∈ D R ( z ) , implying the holomorphy of e ′ ○ f by [25, Theorem 1.7.6, p. 48-49].‘ e )( i ) ⇒ e )( ii ) ⇒ e )( iii ) ⇒ e )( i ) ’: The first implication is obvious, the secondand the third follow from the scalar version of Hartogs’ theorem (see [23, Theorem2.2.8, p. 28]).‘ a )( i ) ⇒ a )( ii ) ⇒ e )( ii ) ’ and ‘ a )( i ) ⇒ a )( iii ) ⇒ e )( iii ) ’: These implications areobvious.‘ a )( i ) ⇒ b ) ⇒ c ) ’: Remark 3.2 b)+c) yields the first implication and the second istrivial.‘ c ) ⇒ e )( i ) ’: This implication holds with G ∶ = E ′ due to E ′ = ( ̂ E ) ′ and the scalarversion of Hartogs’ theorem.‘ e )( i ) ⇒ d ) ’: Let f ∶ Ω → E be such that e ′ ○ f is holomorphic on Ω for every e ′ ∈ G . Then e ′ ○ f ○ φ − is C ∞ on φ ( Ω ) for each e ′ ∈ G and we even obtain f ○ φ − ∈ C ∞ ( φ ( Ω ) , E ) by the weak-strong principle [9, Corollary 10 (a), p. 233].Furthermore, the holomorphy of e ′ ○ f implies ⟨ e ′ , ( ∂ j ) E ( f ○ φ − )( x )⟩ = ( ∂ j ) C ( e ′ ○ ( f ○ φ − ))( x ) = , x ∈ φ ( Ω ) , e ′ ∈ G, for every ≤ j ≤ d by [23, Definition 2.1.1, p. 23]. We conclude that d ) is valid since G is separating.‘ d ) ⇒ e )( i ) ’: Let f ∈ C ∞ R ( Ω , E ) and ∂ j ( f ○ φ − ) = for all ≤ j ≤ d . Then f ○ φ − isweakly C ∞ on φ ( Ω ) and ( ∂ j ) C ( e ′ ○ ( f ○ φ − ))( x ) = ⟨ e ′ , ( ∂ j ) E ( f ○ φ − )( x )⟩ = , x ∈ φ ( Ω ) , e ′ ∈ E ′ , for every ≤ j ≤ d . We derive e )( i ) with G ∶ = E ′ from the scalar version of Hartogs’theorem and [23, Definition 2.1.1, p. 23].‘ a )( i ) ⇒ f )( i ) ’: A consequence of Remark 3.2 b)+d) with K ∶ = ∅ and G ∶ = E ′ .‘ f )( i ) ⇔ f )( ii ) ⇔ f )( iii ) ’: Follows from the correponding equivalences in case e).‘ f )( i ) ⇒ c ) ’: Fix ≤ j ≤ d and z ∈ Ω and consider the map f ○ π j,z ∶ π − j,z ( Ω ) → E .Let w ∈ π − j,z ( Ω ) ∖ π − j,z ( K ) . Then π j,z ( w ) ∈ Ω ∖ K and thus there is a neighbourhood U ⊂ Ω of π j,z ( w ) such that f is bounded on U implying that f ○ π j,z is bounded onthe neighbourhood π − j,z ( U ) of w . Thus we can apply [19, 5.2 Theorem, p. 35] to f ○ π j,z and obtain that ( f ○ π z,j ) ′ ( w ) exists in E for all w ∈ π − j,z ( Ω ) implying for w = z j ∂ e j C f ( z ) = ( f ○ π z,j ) ′ ( z j ) ∈ E. (cid:3) The preceding theorem generalises corresponding theorems for E -valued holo-morphic functions in one variable given in [27, Satz 10.11, p. 241] for quasi-complete E , in [21, Théorème 1, p. 37-38] (cf. [24, 16.7.2 Theorem, p. 362-363]) for E withccp and more general in [19, 2.1 Theorem and Definition, p. 17-18] and [19, 5.2Theorem, p. 35] for locally complete E . In several variables our theorem improves[7, Theorem 3.2, p. 83-84] where E has to be sequentially complete and even in onevariable it is more general than the mentioned ones due to Theorem 3.20 c). Theequivalence ‘ a )( i ) ⇔ a )( iii ) ’ is Hartogs’ theorem and can be found for Banach-valued holomorphic functions on an open set Ω ⊂ C d in [34, 36.1 Theorem, p.265] and for holomorphic functions with values in a sequentially complete space in[7, Corollary 3.6, p. 85]. The equivalence ‘ a )( i ) ⇔ e )( i ) ’ is also contained in [9,Corollary 10 (a), p. 233].The following two corollaries improve [7, Corollary 3.7, p. 85] from sequentiallycomplete E to locally complete E . OLOMORPHIC 19
Corollary.
Let E be a locally complete lcHs over C and Ω ⊂ C d open. If f ∶ Ω → E is holomorphic, then ∂ β C f is holomorphic for all β ∈ N d .Proof. Let β ∈ N d and ≤ j ≤ d . Then we deduce from Corollary 3.13 and Cauchy’sintegral formula that ∂ e j C ( ∂ β C f )( z ) = ∂ β + e j C f ( z ) ∈ E, z ∈ Ω . It follows from Theorem 3.20 ‘ a ) ⇔ c ) ’ that ∂ β C f is holomorphic. (cid:3) Defining the subspace C ∞ C ( Ω , E ) of D ∞ C ( Ω , E ) which consists of all elements of D ∞ C ( Ω , E ) such that all complex partial derivatives of any order are continuous, westate the following consequence of the corollary above.3.22. Corollary.
Let E be a locally complete lcHs over C and Ω ⊂ C d open. Thenthe following statements are equivalent for a function f ∶ Ω → E .a) f is holomorphic on Ω .b) f ∈ C ∞ C ( Ω , E ) .Proof. The implication ‘ b ) ⇒ a ) ’ follows from Theorem 3.20 ‘ c ) ⇒ a ) ’. The otherimplication is a consequence of Corollary 3.13 and Corollary 3.21 which gives thatthe holomorphic function ∂ β C f , β ∈ N d , is continuous by Remark 3.2 b). (cid:3) The following generalisation of Proposition 3.16 describes the relation betweenhigher order real and complex partial derivatives. For convenience we recall thedefinition of higher real partial derivatives. Let k ∈ N , ∞ , Ω ⊂ R d open, E anlcHs and f ∈ C k ( Ω , E ) . For β = ( β , . . . , β d ) ∈ N d with ∣ β ∣ ∶ = ∑ dj = β j ≤ k we set ∂ β j R f ∶ = ( ∂ β j R ) E f ∶ = f if β j = , and ∂ β j R f ∶ = ( ∂ β j R ) E f ∶ = ( ∂ e j R ) ⋯ ( ∂ e j R )´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ β j -times f if β j ≠ as well as ∂ β R f ∶ = ( ∂ β R ) E f ∶ = ( ∂ β R ) ⋯ ( ∂ β d R ) f. Proposition. If E is a locally complete lcHs over C , Ω ⊂ C d open and f ∶ Ω → E holomorphic, then f ∈ C ∞ R ( Ω , E ) and ∂ β R ( f ○ φ − )( x ) = i ∑ dk = β k ∂ ( β + β ,...,β d − + β d ) C f ( φ − ( x )) , x ∈ φ ( Ω ) , β ∈ N d . (11) Proof.
Due to Theorem 3.20 ‘ a ) ⇔ d ) ’ we have f ∈ C ∞ R ( Ω , E ) implying that theleft-hand side of (11) is defined whereas the right-hand side is defined by Cauchy’sintegral formula. Now, for β = equation (11) is trivial. Let n ∈ N and assumethat (11) holds for all β ∈ N d with ∣ β ∣ = n . Let β ∈ N d with ∣ β ∣ = n + . Then thereis j ∈ N , ≤ j ≤ d , and ̃ β ∈ N d with ∣̃ β ∣ = n such that β = ̃ β + e j . We set g ∶ = ∂ (̃ β + ̃ β ,..., ̃ β d − + ̃ β d ) C f and C ∶ = i ∑ dk = ̃ β k and obtain for x ∈ φ ( Ω ) by Schwarz’ theorem ∂ β R ( f ○ φ − )( x ) = ∂ e j R [ ∂ ̃ β R ( f ○ φ − )]( x ) = C ⋅ ∂ e j R ( g ○ φ − )( x ) . We deduce from Corollary 3.21 that g is holomorphic and from Corollary 3.19 that g ∈ O ( Ω , E ) . Thus we can apply Proposition 3.16 to g and compute ∂ e j R ( g ○ φ − )( x ) = i∂ e j C g ( φ − ( x )) , if j is even, and ∂ e j R ( g ○ φ − )( x ) = ∂ e j C g ( φ − ( x )) , if j is odd, which implies by Corollary 3.13 ∂ β R ( f ○ φ − )( x ) = C ⋅ ∂ e j R ( g ○ φ − )( x ) = i ∑ dk = β k ∂ β C f ( φ − ( x )) . (cid:3) The identity theorem for vector-valued holomorphic functions in several variablestakes the following form where the vector-valued one variable case can be found in [9,Corollary 10 (c), p. 233] and the scalar-valued several variables case for example in[25, Proposition 1.7.10, p. 50]. Its version for Banach-valued holomorphic functionson an open subset of a Banach space is given in [34, 5.7 Proposition, p. 37].3.24.
Theorem (Identity theorem) . Let E be a locally complete lcHs over C , F ⊂ E a locally closed subspace, Ω ⊂ C d open and connected and f ∶ Ω → E holomorphic. If(i) the set Ω F ∶ = { z ∈ Ω ∣ f ( z ) ∈ F } has an accumulation point in Ω , or if(ii) there exists z ∈ Ω such that ∂ β C f ( z ) ∈ F for all β ∈ N d ,then f ( z ) ∈ F for all z ∈ Ω .Proof. This follows from Proposition 3.17 and [9, Corollary 8, p. 232] with theFréchet-Schwartz space Y ∶ = O ( Ω ) and the separating subspace X ∶ = span { δ z ∣ z ∈ Ω F } ⊂ Y ′ in (i) resp. X ∶ = span { δ z ○ ∂ β C ∣ β ∈ N d } ⊂ Y ′ in (ii). (cid:3) For the definition of local closedness see [35, Defintion 5.1.14, p. 154-155]. Inparticular, every locally complete subspace of E is locally closed by [35, Proposition5.1.20 (i), p. 155].3.25. Theorem (Liouville) . Let E be a locally complete lcHs over C , f ∶ C d → E holomorphic and k ∈ N . Then the following assertions are equivalent.a) f is a polynomial of degree ≤ k .b) ∀ α ∈ A ∃ C, R > ∀ z ∈ C d , ∣ z ∣ ≥ R ∶ p α ( f ( z )) ≤ C ∣ z ∣ k Proof.
The implication ‘ a ) ⇒ b ) ’ is obvious and the converse implication holdsdue to the power series expansion Theorem 3.18 of f around zero and the Cauchyinequality (7). (cid:3) Let Ω ⊂ C d open and connected. A set A ⊂ Ω is called thin if for every z ∈ A there are R > with D R ( z ) ⊂ Ω and f ∈ O ( D R ( z )) , f ≠ , such that that f = on A ∩ D R ( z ) (see e.g. [22, Chap. 1, Sec. C, 1. Definition, p. 19]). A thin set A ⊂ Ω is nowhere dense by [22, p. 19] and thus the complement Ω ∖ A contains a dense opensubset.3.26. Theorem (Riemann’s removable singularities theorem) . Let E be an lcHsover C , G ⊂ E ′ a subspace, Ω ⊂ C d open and connected, A ⊂ Ω thin and closed and f ∶ Ω ∖ A → E holomorphic. If for every z ∈ Ω there is a polydisc D R ( z ) ⊂ Ω suchthat f is bounded on D R ( z ) ∖ A and(i) G is separating and E B r -complete, or if(ii) G is dense in E ′ b and E locally complete,then f extends holomorphically to Ω .Proof. First, we remark that e ′ ○ f is holomorphic and bounded on D R ( z ) ∖ A forsome R and each z ∈ Ω and e ′ ∈ G . Due to the scalar version of Riemann’s removablesingularities theorem (see [22, Chap. 1, Sec. C, 3. Theorem, p. 19]) e ′ ○ f extends toa holomorphic function f e ′ on Ω for each e ′ ∈ G . Let ( Ω n ) n ∈ N be any exhaustion of Ω with relatively compact, open and connected sets such that Ω n ⊂ Ω n + for every n ∈ N . Since M ∶ = Ω ∖ A is dense in Ω , we have ∂ Ω n ⊂ Ω n + = A ∩ Ω n + . Hence ourstatement is true by [9, Corollary 18, p. 238] with F ( Ω ) ∶ = O ( Ω ) . (cid:3) OLOMORPHIC 21 B r -complete spaces (see [24, p. 183]) are also called infra-Pták spaces and, forinstance, every Fréchet space is B r -complete by [24, 9.5.2 Krein-˘Smulian Theorem,p. 184]. Condition (ii) is especially fulfilled if G is separating and E semireflexive(see [9, p. 234]).We close our treatment of holomorphic functions with another application of thepower series expansion given in Theorem 3.18. For an lcHs E over C , z ∈ C d and R ∈ ( , ∞ ) d we set A ( D R ( z ) , E ) ∶ = O ( D R ( z ) , E ) ∩ C ( D R ( z ) , E ) and equip this space with the system of seminorms generated by ∥ f ∥ α ∶ = sup w ∈ D R ( z ) p α ( f ( w )) , f ∈ A ( D R ( z ) , E ) , for α ∈ A . We write A ( D R ( z )) ∶ = A ( D R ( z ) , C ) and in the case z = and R = ( , . . . , ) this space is known as the polydisc algebra . Further, we denote by P ( D R ( z ) , E ) the space of E -valued polynomials on D R ( z ) and aim to prove thatthe E -valued polynomials are dense in A ( D R ( z ) , E ) if E is locally complete. If E is a Fréchet space, this result can be found in [5, 9. Corollary, p. 5]. Our proof isalong the lines of the one in the case E = C , z = , d = and R = given in [36, p.366].3.27. Corollary.
Let E be a locally complete lcHs over C , z ∈ C d and R ∈ ( , ∞ ) d .Then the following statements hold.a) The tensor product P ( D R ( z ) , C ) ⊗ E is dense in A ( D R ( z ) , E ) .b) If E is complete, then A ( D R ( z ) , E ) ≅ A ( D R ( z )) εE ≅ A ( D R ( z ))̂ ⊗ ε E where ≅ stands for topologically isomorphic and A ( D R ( z ))̂ ⊗ ε E is the com-pletion of the injective tensor product A ( D R ( z )) ⊗ ε E .c) A ( D R ( z )) has the approximation property.Proof. The map S ∶ A ( D R ( z )) εE → A ( D R ( z ) , E ) , u z→ [ w ↦ u ( δ w )] , is a (topological) isomorphism into, i.e. to its range, by [6, 3.1 Bemerkung, p.141] with Y ∶ = A ( D R ( z )) and Theorem 3.20 ‘ a ) ⇔ e ) ’. Let α ∈ A , ε > and f ∈ A ( D R ( z ) , E ) . Since D R ( z ) is compact, f is uniformly continuous on D R ( z ) andthus there is δ > such that p α ( f ( w ) − f ( x )) < ε for all x, w ∈ D R ( z ) with ∣ w − x ∣ < δ .Choosing r > such that − δ √ d max ≤ j ≤ d ( R j + ∣ z j ∣) < r < , we get / r > and thus D R ( z ) ⊂ D R / r ( z ) . Furthermore, for every w ∈ D R ( z ) wehave ∣ w ∣ ≤ √ d max ≤ j ≤ d ∣ w j ∣ ≤ √ d max ≤ j ≤ d (∣ w j − z j ∣ + ∣ z j ∣) ≤ √ d max ≤ j ≤ d ( R j + ∣ z j ∣) connoting ∣ w − rw ∣ = ( − r )∣ w ∣ < δ √ d max ≤ j ≤ d ( R j + ∣ z j ∣) ∣ w ∣ ≤ δ. Therefore sup w ∈ D R ( z ) p α ( f ( w ) − f ( rw )) ≤ ε (12) and we set g ∶ D R / r ( z ) → E, g ( w ) ∶ = f ( rw ) , which is function in O ( D R / r ( z ) , E ) . ByTheorem 3.18 there is N ∈ N such that for all n ≥ N ∥ g − ∑ ∣ β ∣ ≤ n ( ⋅ − z ) β ⊗ ( ∂ β C ) E f ( z ) β ! ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ = ∶ T n ∥ α = sup w ∈ D R ( z ) p α ( g ( w ) − ∑ ∣ β ∣ ≤ n ( ∂ β C ) E f ( z ) β ! ( w − z ) β ) < ε. We observe that the restriction of T N to D R ( z ) is an element of P ( D R ( z ) , C ) ⊗ E = P ( D R ( z ) , E ) and thus of A ( D R ( z )) ⊗ E as well. We conclude from (12) that ∥ f − T N ∥ α ≤ ∥ f − g ∥ α + ∥ g − T N ∥ α < ε which proves our first statement. The second follows from the first by [31, 3.5Remark, p. 7] because A ( D R ( z )) is a Banach space and thus complete. The laststatement results from the second by [24, 18.1.8 Theorem, p. 400]. (cid:3) References [1] W. Arendt and N. Nikolski. Vector-valued holomorphic functions revisited.
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TU Hamburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, Gebäude E,21073 Hamburg, Germany
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