In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces (or Fréchet spaces more generally), typically of infinite dimension. It is one aspect of nonlinear functional analysis.
A first step in extending the theory of holomorphic functions beyond one complex dimension is considering so-called vector-valued holomorphic functions, which are still defined in the complex plane C, but take values in a Banach space. Such functions are important, for example, in constructing the holomorphic functional calculus for bounded linear operators.
Definition. A function f : U → X, where U ⊂ C is an open subset and X is a complex Banach space, is called holomorphic if it is complex-differentiable; that is, for each point z ∈ U the following limit exists:f'(z)=\lim\zeta\to
f(\zeta)-f(z) \zeta-z .
One may define the line integral of a vector-valued holomorphic function f : U → X along a rectifiable curve γ : [''a'', ''b''] → U in the same way as for complex-valued holomorphic functions, as the limit of sums of the form
\sum1f(\gamma(tk))(\gamma(tk)-\gamma(tk-1))
where a = t0 < t1 < ... < tn = b is a subdivision of the interval [''a'', ''b''], as the lengths of the subdivision intervals approach zero.
It is a quick check that the Cauchy integral theorem also holds for vector-valued holomorphic functions. Indeed, if f : U → X is such a function and T : X → C a bounded linear functional, one can show that
T\left(\int\gammaf(z)dz\right)=\int\gamma(T\circf)(z)dz.
\int\gammaf(z)dz=0
which proves the Cauchy integral theorem in the vector-valued case.
Using this powerful tool one may then prove Cauchy's integral formula, and, just like in the classical case, that any vector-valued holomorphic function is analytic.
A useful criterion for a function f : U → X to be holomorphic is that T o f : U → C is a holomorphic complex-valued function for every continuous linear functional T : X → C. Such an f is weakly 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 complex Banach spaces X and Y and an open set U ⊂ X, f : U → Y is called holomorphic if the Fréchet derivative of f exists at every point in U. 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 a power 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.
In general, given two complex topological vector spaces X and Y and an open set U ⊂ X, there are various ways of defining holomorphy of a function f : U → Y. Unlike the finite dimensional setting, when X and Y 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 when X and Y are locally 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 spaces X and Y satisfy some additional constraints.
Gateaux holomorphy is the direct generalization of weak holomorphy to the fully infinite dimensional setting.
Let X and Y be locally convex topological vector spaces, and U ⊂ X an open set. A function f : U → Y is said to be Gâteaux holomorphic if, for every a ∈ U and b ∈ X, and every continuous linear functional φ : Y → C, the function
f\varphi(z)=\varphi\circf(a+zb)
In the analysis of Gateaux holomorphic functions, any properties of finite-dimensional holomorphic functions hold on finite-dimensional subspaces of X. 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.
| infty | |
| f(x+y)=\sum | |
| n=0 |
| 1 | |
| n! |
\widehat{D}nf(x)(y)
Here,
\widehat{D}nf(x)(y)
If f : (U ⊂ X1) × (V ⊂ X2) → Y is a function which is separately Gateaux holomorphic in each of its arguments, then f is Gateaux holomorphic on the product space.
A function f : (U ⊂ X) → Y is hypoanalytic if f ∈ HG(U,Y) and in addition f is continuous on relatively compact subsets of U.
A function f ∈ HG(U,Y) is holomorphic if, for every x ∈ U, the Taylor series expansion
| infty | |
| f(x+y)=\sum | |
| n=0 |
| 1 | |
| n! |
\widehat{D}nf(x)(y)
A function f : (U ⊂ X) → Y is said to be locally bounded if each point of U has a neighborhood whose image under f is bounded in Y. If, in addition, f is Gateaux holomorphic on U, then f is locally bounded holomorphic. In this case, we write f ∈ HLB(U,Y).