#### Thesis Title

### Elements of Functional Analysis and Applications

#### Date of Graduation

Summer 2019

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Shouchuan Hu

#### Abstract

Functional analysis is a branch of mathematical analysis that studies vector spaces with a limit structure (such as a norm or inner product), and functions or operators defined on these spaces. Functional analysis provides a useful framework and abstract approach for some applied problems in variety of disciplines. In this thesis, we will focus on some basic concepts and abstract results in functional analysis, and then demonstrate their power and relevance by solving some applied problems under the framework. We will give the definitions and provide some examples of some different spaces (such as metric spaces, normed spaces and inner product spaces), and some useful operators (such as differential operators, integral operators, contractions, completely continuous operators, etc.). Specifically, we will use the Banach Fixed Point Theory to prove the Existence and Uniqueness of the solutions to an Initial Value Problem for a system of n-first order differential equations. Then we will use the Leray-Schauder Fixed Point Theory to prove the existence of solutions for a nonlinear Sturm-Liouville Boundary Value Problem. Finally, we will demonstrate how the maximum-area problem can be proved handedly by the Calculus of Variation approach, a subset of Functional Analysis.

#### Keywords

space, operator, functional, fixed point theory, Initial Value Problem, Boundary Value Problem, max area problem

#### Subject Categories

Analysis

#### Copyright

© Chengting Yin

#### Recommended Citation

Yin, Chengting, "Elements of Functional Analysis and Applications" (2019). *MSU Graduate Theses*. 3411.

https://bearworks.missouristate.edu/theses/3411