Inmathematics,infinite-dimensional holomorphy is a branch offunctional analysis. It is concerned with generalizations of the concept ofholomorphic function to functions defined and taking values incomplexBanach spaces (orFréchet spaces more generally), typically of infinite dimension. It is one aspect ofnonlinear functional analysis.

A first step in extending the theory of holomorphic functions beyond one complex dimension is considering so-calledvector-valued holomorphic functions, which are still defined in thecomplex planeC, but take values in a Banach space. Such functions are important, for example, in constructing theholomorphic functional calculus forbounded linear operators.

Definition. A functionf :U →X, whereU ⊂C is anopen subset andX is a complex Banach space, is calledholomorphic if it is complex-differentiable; that is, for each pointz ∈U the followinglimit exists:

${\displaystyle f^{\prime }(z)=\underset{\zeta \to z}{lim}\frac{f(\zeta )-f(z)}{\zeta -z}.}$

One may define theline integral of a vector-valued holomorphic functionf :U →X along arectifiable curve γ : [a,b] →U in the same way as for complex-valued holomorphic functions, as the limit of sums of the form

${\displaystyle \sum _{1\le k\le n}f(\gamma (t_k))(\gamma (t_k)-\gamma (t_{k-1}))}$

wherea =t₀ <t₁ < ... <t_n =b is a subdivision of the interval [a,b], as the lengths of the subdivision intervals approach zero.

It is a quick check that theCauchy integral theorem also holds for vector-valued holomorphic functions. Indeed, iff :U →X is such a function andT :X →C a bounded linear functional, one can show that

${\displaystyle T\left({\int }_{\gamma }f(z)dz\right)={\int }_{\gamma }(T\circ f)(z)dz.}$

Moreover, thecompositionTof :U →C is a complex-valued holomorphic function. Therefore, for γ asimple closed curve whose interior is contained inU, the integral on the right is zero, by the classical Cauchy integral theorem. Then, sinceT is arbitrary, it follows from theHahn–Banach theorem that

${\displaystyle {\int }_{\gamma }f(z)dz=0}$

which proves the Cauchy integral theorem in the vector-valued case.

Using this powerful tool one may then proveCauchy's integral formula, and, just like in the classical case, that any vector-valued holomorphic function isanalytic.

A useful criterion for a functionf :U →X to be holomorphic is thatTof :U →C is a holomorphic complex-valued function for everycontinuous linear functionalT :X →C. Such anf isweakly holomorphic. It can be shown that a function defined on an open subset of the complex plane with values in a Fréchet space is holomorphic if, and only if, it is weakly holomorphic.

More generally, given two complexBanach spacesX andY and an open setU ⊂X,f :U →Y is calledholomorphic if theFréchet derivative off exists at every point inU. One can show that, in this more general context, it is still true that a holomorphic function is analytic, that is, it can be locally expanded in apower series. It is no longer true however that if a function is defined and holomorphic in a ball, its power series around the center of the ball is convergent in the entire ball; for example, there exist holomorphic functions defined on the entire space which have a finite radius of convergence.^[1]

In general, given two complextopological vector spacesX andY and an open setU ⊂X, there are various ways of defining holomorphy of a functionf :U →Y. Unlike the finite dimensional setting, whenX andY are infinite dimensional, the properties of holomorphic functions may depend on which definition is chosen. To restrict the number of possibilities we must consider, we shall only discuss holomorphy in the case whenX andY arelocally convex.

This section presents a list of definitions, proceeding from the weakest notion to the strongest notion. It concludes with a discussion of some theorems relating these definitions when the spacesX andY satisfy some additional constraints.

Gateaux holomorphy is the direct generalization of weak holomorphy to the fully infinite dimensional setting.

LetX andY be locally convex topological vector spaces, andU ⊂X an open set. A functionf :U →Y is said to beGâteaux holomorphic if, for everya ∈U andb ∈X, and everycontinuous linear functional φ :Y →C, the function

${\displaystyle f_{\phi }(z)=\phi \circ f(a+zb)}$

is a holomorphic function ofz in a neighborhood of the origin. The collection of Gâteaux holomorphic functions is denoted by H_G(U,Y).

In the analysis of Gateaux holomorphic functions, any properties of finite-dimensional holomorphic functions hold on finite-dimensional subspaces ofX. However, as usual in functional analysis, these properties may not piece together uniformly to yield any corresponding properties of these functions on full open sets.

• Iff ∈U, thenf hasGateaux derivatives of all orders, since forx ∈U andh₁, ...,h_k ∈X, thek-th order Gateaux derivativeD^kf(x){h₁, ...,h_k} involves only iterated directional derivatives in the span of theh_i, which is a finite-dimensional space. In this case, the iterated Gateaux derivatives are multilinear in theh_i, but will in general fail to be continuous when regarded over the whole spaceX.
• Furthermore, a version of Taylor's theorem holds:

${\displaystyle f(x+y)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\hat{D}}^{n}f(x)(y)}$

Here,${\displaystyle {\hat{D}}^{n}f(x)(y)}$ is thehomogeneous polynomial of degreen iny associated with themultilinear operatorDⁿf(x). The convergence of this series is not uniform. More precisely, ifV ⊂X is afixed finite-dimensional subspace, then the series converges uniformly on sufficiently small compact neighborhoods of 0 ∈Y. However, if the subspaceV is permitted to vary, then the convergence fails: it will in general fail to be uniform with respect to this variation. Note that this is in sharp contrast with the finite dimensional case.

• Hartog's theorem holds for Gateaux holomorphic functions in the following sense:

Iff : (U ⊂X₁) × (V ⊂X₂) →Y is a function which isseparately Gateaux holomorphic in each of its arguments, thenf is Gateaux holomorphic on the product space.

A functionf : (U ⊂X) →Y ishypoanalytic iff ∈H_G(U,Y) and in additionf is continuous onrelatively compact subsets ofU.

A functionf ∈ H_G(U,Y) isholomorphic if, for everyx ∈U, theTaylor series expansion

${\displaystyle f(x+y)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\hat{D}}^{n}f(x)(y)}$

(which is already guaranteed to exist by Gateaux holomorphy) converges and is continuous fory in a neighborhood of 0 ∈X. Thus holomorphy combines the notion of weak holomorphy with the convergence of the power series expansion. The collection of holomorphic functions is denoted by H(U,Y).

A functionf : (U ⊂X) →Y is said to belocally bounded if each point ofU has a neighborhood whose image underf is bounded inY. If, in addition,f is Gateaux holomorphic onU, thenf islocally bounded holomorphic. In this case, we writef ∈ H_LB(U,Y).

• Richard V. Kadison, John R. Ringrose,Fundamentals of the Theory of Operator Algebras, Vol. 1: Elementary theory. American Mathematical Society, 1997.ISBN 0-8218-0819-2. (See Sect. 3.3.)
• Soo Bong Chae,Holomorphy and Calculus in Normed Spaces, Marcel Dekker, 1985.ISBN 0-8247-7231-8.
• ^ Lawrence A. Harris,Fixed Point Theorems for Infinite Dimensional Holomorphic Functions (undated).

• Banach spaces