A Holomorphic Map in Infinite Dimensions

We prove, in great detail, holomorphy E ⊓ C (I ,Π ) → C (I ,Π ) of the map (x , y) 7→ x ◦ [ id , y ] where [ id , y ] : I ∋ t 7→ (t , y (t)) for a real compact interval I , and where Π is a complex Banach space and E is a certain locally convex space of continuous functions x : I× (υsΠ ) → υsΠ for which x (t , ·) is holomorphic for all t ∈ I . We also discuss aspects of the application of this result to establishing a holomorphic solution map (ξ , φ) 7→ y for functions y : I → υsΠ satisfying the ordinary differential equation y ′ = φ ◦ [ id , y ] with initial condition y (t0) = ξ . In [5] , the following problem was considered. Fix a vector ξ0 in a given complex Banach space Π , and let B0 be an open ball in Π centered at ξ0 . Consider differentiable curves y in B0 defined on the compact real interval I = [ t0 − A , t0 + A ] and satisfying the differential equation E(x, y) : y (t) = x(t , y (t)) for all t ∈ I with initial condition y (t0) = ξ0 , where x is a suitable function defined on I ×B0 and having values in Π . Theorem 2 in [5, p. 85 ] gave the result that if x0 is suitably small in a certain Banach space E1 of functions x , then an open neighbourhood U of x0 in E1 exists such that for all x ∈ U there is a unique y with E(x, y) , and the function U ∋ x 7→ y defines a holomorphic map E1 → C (I ,Π ) . See Constructions 3 below and the discussion next to them for the precise formulation. It is essential for the preceding result that the map (x, y) 7→ x ◦ [ id , y ] : I ∋ t 7→ x(t , y (t)) be holomorphic. The purpose of this note is to establish this in a setting more general than the one considered in [5] , see Theorem 2 below. We also indicate the main steps of the proof of Theorem 4 below generalizing [5, Theorem 2 ] . We use the conventions of [2] . Therefrom, in particular, we recall that R and C are the standard real and complex topological fields, respectively. The topology of a topological vector space E is τ rd E , and its underlying set is υsE . Its filter of zero neighbourhoods is No E , and BsE is the set of bounded sets. The class of complex Banach(able topological vector) spaces is BaS(C) . We also recall from [2, Section 3 ] that the particular holomorphy class H T has as its members exactly the maps f̃ = (E ,F , f ) of complex Hausdorff locally convex spaces E ,F where we have F locally (= Mackey, see [3, p. 196 ] or [4, Lemma 2.2, p. 15 ]) complete, and f a function with dom f ∈ τ rd E , i.e. open set in E , and rng f ⊆ υsF , and f continuous τrdE → τrdF and f̃ directionally differentiable, the last one meaning that for all fixed x ∈ dom f and u ∈ υsE the limit δf `(x, u) of t(f `(x+ t u)− f `x) as CI \{0} ∋ t → 0 exists in the space F . Recall that we let f `x be the function value of f at x , and that f [A ] = f `̀A = { y : ∃x ∈ A ; (x, y) ∈ f } . In [2] , we also agreed on the definitions f = {(y , x) : (x, y) ∈ f } and dom f = { x : ∃ y ; (x, y) ∈ f } and dom2f = dom (dom f ) . 2000 Mathematics Subject Classification 46G20.