Introduction to normed and Hilbert spaces: Intended to the students of the third year Bachelor in Mathematics

Abstract

This manuscript provides an introductory exploration of the fundamental concepts and properties of normed and linear spaces. It is intended for thirdyear Bachelor of Mathematics students. It is divided into several chapters and sections, each covering specific topics. It also includes exercises to reinforce understanding of the concepts presented. Normed and linear spaces are essential mathematical structures used in various fields such as functional analysis, linear algebra, and mathematical physics. This manuscript serves as a comprehensive guide for students to develop a solid understanding of these spaces, their properties, and their applications. By studying this subject, students can lay a strong foundation for advanced mathematical studies and applications in various scientific disciplines. Chapter one introduces the basic concepts of metric spaces. It covers definitions and examples of metric spaces, open and closed balls, convergence, continuity, and open and closed sets. Chapter two focuses on normed linear spaces. It discusses norms over E, distance associated with a norm, properties of the norm, equivalent norms, and Banach spaces. It also explores finite- dimensional normed linear spaces, continuous linear applications, and the product of two normed spaces. The third chapter delves into Hilbert spaces, which are special types of normed linear spaces. It introduces the scalar product, orthogonality, the Hilbert projection theorem, the theorem of F. Riesz, and orthonormal systems in a Hilbert space. Overall, this manuscript provides a systematic and comprehensive introduction to normed and linear spaces. It covers essential concepts, definitions, and properties, allowing students to develop a solid foundation in this area