Comparative Study of Several Bases in Functional Analysis

Sammanfattning: From the beginning of the study of spaces in functional analysis, bases have been an indispensable tool for operating with vectors and functions over a concrete space. Bases can be organized by types, depending on their properties. This thesis is intended to give an overview of some bases and their relations. We study Hamel basis, Schauder basis and Orthonormal basis; we give some properties and compare them in different spaces, explaining the results. For example, an infinite dimensional Hilbert space will never have a basis which is a Schauder basis and a Hamel basis at the same time, but if this space is separable it has an orthonormal basis, which is also a Schauder basis. The project deals mainly with Banach spaces, but we also talk about the case when the space is a pre Hilbert space.