Weierstrass�s Global Division Theorem and Continuity of Linear Operators in H-spaces

Abstract

We introduce here new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Author has introduced this concepts in 2002 but progress in different areas of mathematics (algebraic geometry, differential equations, category theory, ets) defined the need to expand fundamental concepts. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Frhet spaces and is stable under the operations of forming countable inductive and projective limits, closed subspaces and factor-spaces. Besides, for H-space the strengthened variant of the closed graph theorem holds true. In the present article generalization of Weierstrasss preparation theorem and the division theorem for germs of holomorphic functions at a point of n-dimensional complex space are considered. The author formulates the global theorem about division in terms of existence and continuity of the linear operator.