Holomorphic Functions Between Banach Spaces
More generally, given two Banach spaces X and Y over the complex numbers and an open set U in 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.
Read more about this topic: Infinite-dimensional Holomorphy
Famous quotes containing the words functions and/or spaces:
“Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
—Henry David Thoreau (1817–1862)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)