Academic Journal of Mathematical Sciences, 2022, 3(1); doi: 10.25236/AJMS.2022.030102.

## Rolle Theorem and Radon-Nikodym Theorem in Bochner Cost

Author(s)

Chenyuan Tian1, Daohong Liu2

Corresponding Author:
Chenyuan Tian
Affiliation(s)

1College of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan, Hubei, China

2College of Mathematics and Physics, Chengdu University of Technology, Chengdu, Sichuan, China

### Abstract

This paper are divided into two mains parts with different affairs to discuss. The first part are some history of how the modern strict mathmaticala analysis built up. And there’s a simple counterexample for Rolle theorem. As we all known Rolle theorem has three requirement to work out. So, there’s some counterexamples for which there’s only two requirements exist, and what is going to happen. The second part gives two counterexamples on Radon-Nikodym theorem and Riesz theorem in the meaning of vector measure. Then there’s a proof for the equivalent of the Radon-Nikodym theorem and Riesz theorem in the meaning of vector measure. Some recent research are given at the last, which means it’s still a vigor field nowadays.

### Keywords

Rolle Theorem, Radon-Nikodym Theorem, Riezs Theorem, Bochner integral

### Cite This Paper

Chenyuan Tian, Daohong Liu. Rolle Theorem and Radon-Nikodym Theorem in Bochner Cost. Academic Journal of Mathematical Sciences (2022) Vol. 3, Issue 1: 8-14. https://doi.org/10.25236/AJMS.2022.030102.

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