Hilbert Spaces and Orthonormal Systems

Abstract

Historically, the theory of Hilbert spaces originated from David Hilbert’s (1862–1943) work on quadratic forms in infinitely many variables with their applications to integral equations. During the period of 1904–1910, Hilbert published a series of six papers, subsequently collected in his classic book Grundzüge einer allemeinen Theorie der linearen lntegralgleichungen published in 1912. It contained many general ideas including Hilbert spaces (ℓ² and L²), the compact operators, and orthogonality, and had a tremendous influence on mathematical analysis and its applications. After many years, John von Neumann (1903–1957) first formulated an axiomatic approach to Hilbert space and developed the modern theory of operators on Hilbert spaces. His remarkable contribution to this area has provided the mathematical foundation of quantum mechanics. Von Neumann’s work has also provided an almost definite physical interpretation of quantum mechanics in terms of abstract relations in an infinite dimensional Hilbert space. It was shown that observables of a physical system can be represented by linear symmetric operators in a Hilbert space, and the eigenvalues and eigenfunctions of the particular operator that represents energy are energy levels of an electron in an atom and corresponding stationary states of the system. The differences in two eigenvalues represent the frequencies of the emitted quantum of light and thus define the radiation spectrum of the substance.