Hilbert Space

Abstract

This chapter develops the basics of the theory of bounded and unbounded linear operators on Hilbert spaces. We will finally construct the spectral decomposition for an unbounded self-adjoint linear operator on a Hilbert space and prove Stone’s Theorem. This shows that strict finitism is in principle sufficient for the basic applications in classical quantum mechanics. This chapter again takes many ideas from Bishop and Bridges [6], and we have to make many changes as well. In particular, the basic definition of linear space has to be modified to fit into strict finitism. The development of the theory of unbounded linear operators on Hilbert spaces follows the ideas in Ye [40, 41], with necessary improvements to fit into strict finitism.