Three topics in analysis: (I) The fundamental theorem of calculus implies that of algebra, (II) Mini sums for the Riesz representing measure, and (III) Holomorphic domination and complex Banach manifolds similar to Stein manifolds.

摘要

We look at three distinct topics in analysis. In the first we give a direct and easy proof that the usual Newton–Leibniz rule implies the fundamental theorem of algebra that any nonconstant complex polynomial of one complex variable has a complex root. Next, we look at the Riesz representation theorem and show that the Riesz representing measure often can be given in the form of mini sums just like in the case of the usual Lebesgue measure on a cube. Lastly, we look at the idea of holomorphic domination and use it to define a class of complex Banach manifolds that is similar in nature and definition to the class of Stein manifolds.;INDEX WORDS: Fundamental theorem of calculus, Fundamental theorem of algebra, Riesz representation theorem, Regular measure, Holomorphic domination, Complex Banach manifolds, Stein manifolds.