Vector-valued holomorphic functions in several variables

K. Kruse
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引用次数: 4

Abstract

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.
多变量的向量值全纯函数
本文给出了局部完全局部凸Hausdorff空间$E$ / $\mathbb{C}$上的若干变量全纯函数的一些民俗定理的显式证明。大多数关于向量值全纯函数的文献都是关于单变量或无穷多变量的情况,而对于(有限多)几个变量的情况则只涉及到或受到类似序列完备性的更强的限制。我们使用的主要工具是柯西积分公式,它是由pettis积分导出的,用于求多变量E值全纯函数的导数。这允许我们推广已知的积分公式,通常使用黎曼积分,从顺序完全$E$到局部完全$E$。我们证明了局部完全空间$E$中值为若干变量的全纯函数的经典定理,包括恒等定理、Liouville定理、Riemann可移动奇点定理和$E$值多盘代数中多项式的密度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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