By Vincent Cossart

ISBN-10: 3540139044

ISBN-13: 9783540139041

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36 Holomorphic Functions of Several Variables 2. 13. Prove that an open subset U of C' is a domain of holomorphy if and only if, for every sequence {z} C U, which converges to a point on the boundary of U, there is an f E R(U) such that {f(z)} is unbounded. 14. Prove that a holomorphic function is an open map (the image of every open subset of the domain is open). 15. -1(U). Prove that, if f vanishes on a non-empty set of the form u n (zo + R'z), then f is identically 0. What must be true of a real linear subspace L C CCn if this result is to hold with Il replaced by L?

In the several variable case we simply apply this result in each variable separately (with the other variables fixed) to conclude that the limit of a sequence of holomorphic functions, converging uniformly on compacta, is holomorphic in each variable separately. Such a limit is also obviously continuous, and so is holomorphic, by Osgood's lemma (we could appeal to Hartog's theorem but we really only need the more elementary Osgood's lemma). A topological vector space, with a topology defined by a sequence of seminorms, as above, and which is complete in this topology, is called a Frechet space.

N, it is differentiable in xj, yj and (9f /azi = 0 in U. A much stronger result is true: A distribution 0, defined in a domain in Can, which satisfies the Cauchy-Riemann equations, a'/azj = 0, j = 1, ... , n in the distribution sense, is actually a holomorphic function. We will not prove this here, but it follows from regularity theorems for elliptic PDE's. 4 Convergence Theorems If U is an open subset of (Cn, then the space of all holomorphic functions on U will be denoted by 7-l(U). This is obviously a complex algebra under the operations of pointwise addition and multiplication of functions.

### Resolution of Surface Singularities by Vincent Cossart

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