Hilbert Space and Applications

Abstract

Functional analysis is one of the most important branches of modern Mathematics and Physics. In functional analysis, Hilbert spaces have a great and positive influence in the development of functional analysis. Hilbert spaces have a rich geometric structure because they are endowed with a scalar product which allows the introduction of the concept of orthogonality of vectors. The aim of this paper is to introduce the Hilbert spaces, and their properties, operations, and applications. We first give an introduction about functional analysis and highlight the importance of Hilbert spaces in the development of functional analysis. Then, we focus on significant and crucial spaces for Hilbert spaces called Banach Spaces. We do so by studying normed spaces and their properties. Finally, we discuss what is meant by Hilbert spaces and what is the relation between Hilbert and Banach spaces.