#### Title

#### Date of Award

2018

#### Document Type

Honors Thesis (Open Access)

#### Department

Colby College. Mathematics and Statistics Dept.

#### Advisor(s)

D. Benjamin Mathes

#### Abstract

There are many instances where the theory of eigenvalues and eigenvectors has its applications. However, Matrix theory, which usually deals with vector spaces with finite dimensions, also has its constraints. Spectral theory, on the other hand, generalizes the ideas of eigenvalues and eigenvectors and applies them to vector spaces with arbitrary dimensions. In the following chapters, we will learn the basics of spectral theory and in particular, we will focus on one of the most important theorems in spectral theory, namely the spectral theorem. There are many different formulations of the spectral theorem and they convey the "same" idea. In this thesis, we are going to see two of the approaches toward the theorem. The intention is that the more aspects you know about the theorem as a tool, the more powerful you are with that tool. Chapter 3;4;5;6 will be devoted to the measure theoretical approach of the spectral theorem and the last chapter will focus on the algebraic approach of the theorem.

#### Keywords

Spectral Theory, The Spectral Theorem, Spectral Radius, Banach Algebra

#### Recommended Citation

Zhang, Muyuan, "On Spectral Theorem" (2018).*Honors Theses.*Paper 900.

https://digitalcommons.colby.edu/honorstheses/900