The open mapping theorem and the fundamental theorem of algebra

Details The open mapping theorem and the fundamental theorem of algebra The open mapping theorem and the fundamental theorem of algebra

2007-10-30

arxiv.org/abs/0710.5648v2

Fixed point theory 9 (2008), 259-266

Mathematics

Complex Variables

7 pages; slight modifications and additions (including references and thanks)

Scientific paper

This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological argument. The proof is elementary in nature and does not use any kind of integration (neither complex nor real). In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems. In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility to achieve it was questioned. We close this note with a simple generalization of the FTA. A short review of certain issues related to the OMT and the FTA is also included.

Affiliated with

Also associated with

No associations

Rating

The open mapping theorem and the fundamental theorem of algebra does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The open mapping theorem and the fundamental theorem of algebra, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The open mapping theorem and the fundamental theorem of algebra will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-365510