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

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  Fréchet  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 Weierstrass’s 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.

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