Vector-valued Holomorphic Functions Defined in The Complex Plane
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.
A function f:U→X defined on an open subset U of the complex plane with values in a complex Banach space X is called holomorphic if it is complex-differentiable; that is, for each point z in U the limit
exists.
One may define the line integral of a vector-valued holomorphic function f:U→X along a rectifiable curve γ → U in the same way as for complex-valued holomorphic functions, as the limit of sums of the form
where a = t0 < t1 < ... < tn = b is a subdivision of the interval, 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
Moreover, the composition Tof:U→C is a complex-valued holomorphic function. Therefore, for γ a simple closed curve whose interior is contained in U, the integral on the right is zero, by the classical Cauchy integral theorem. Then, since T is arbitrary, it follows from the Hahn–Banach theorem that
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.
Read more about this topic: Infinite-dimensional Holomorphy
Famous quotes containing the words functions, defined, complex and/or plane:
“In today’s world parents find themselves at the mercy of a society which imposes pressures and priorities that allow neither time nor place for meaningful activities and relations between children and adults, which downgrade the role of parents and the functions of parenthood, and which prevent the parent from doing things he wants to do as a guide, friend, and companion to his children.”
—Urie Bronfenbrenner (b. 1917)
“As long as the “woman’s work” that some men do is socially devalued, as long as it is defined as woman’s work, as long as it’s tacked onto a “regular” work day, men who share it are likely to develop the same jagged mouth and frazzled hair as the coffee-mug mom. The image of the new man is like the image of the supermom: it obscures the strain.”
—Arlie Hochschild (20th century)
“Specialization is a feature of every complex organization, be it social or natural, a school system, garden, book, or mammalian body.”
—Catharine R. Stimpson (b. 1936)
“As for the dispute about solitude and society, any comparison is impertinent. It is an idling down on the plane at the base of a mountain, instead of climbing steadily to its top.”
—Henry David Thoreau (1817–1862)