A primer of Hilbert space
MIUR (Ministero dell'Istruzione Università e Ricerca)
This book deals with the very basic theory of Hilbert space. Chapter 1 deals with the fundamental theory of vector spaces. The notion of a vector space is recalled, together with related techniques. Key definitions and theorems concerning vector spaces in general and linear independence are then reviewed.
In chapter 2 the notions of a inner product, normed space and metric space are examined, together with their mutual relationships.
Finally, in chapter 3 the very basic tools in the theory of Hilbert space are studied, introducing the natural generalisation of the customary vector space notions to the inﬁnite dimensional framework. In particular, the matter of expanding a vector in terms of a (not necessarily ﬁnite) orthogonal basis is introduced. Several examples, problems and exercises are proposed.
Table of Contents:
Chapter 1. Vector spaces: basic definitions and properties
1. Definition of a vector space
2. Examples of vector spaces
3. Linear independence and dimension of vector spaces
4. Vector space isomorphisms
Inner product, normed and metric spaces
1. Inner product spaces
2. Normed spaces
3. Metric spaces
Chapter 3. Hilbert space
1. Main definitions
2. Examples of Hilbert spaces
3. Separable Hilbert space
4. Convergence in Hilbert space
5. Orthonormal Bases
6. Orthogonal Subspaces and Projections
7. Riesz representation theorem
ISBN 9783862884278. LINCOM Textbooks in Mathematics 01.110pp. 2013.