CHAPTER V - GEOMETRY OF HILBERT SPACE

Abstract

This chapter highlights the geometry of Hilbert space. A linear space H (under multiplication by real numbers) is said to be a (real) Hilbert space if there is a rule that maps each pair of points (vectors) of H onto a real number called the scalar product of the vectors x, y and denoted by (x, y). The well-known expression for the scalar product of vectors in three-dimensional space in terms of their coordinates is relative to an orthogonal coordinate system. The definition of an isomorphism between Hilbert spaces is formulated in analogy with the definitions of equivalence of sets, isometry of metric spaces, and isomorphism of linear spaces. Two Hilbert spaces H′, H″ are said to be isomorphic if there exists between them a one–one correspondence with some properties. Linear operations and a norm can be naturally introduced into the completion of a normed linear space so as to make it again a normed linear space. If H is a Hilbert space, a scalar product can also be naturally introduced into its completion.