The algebra of operators in the space of functions is of considerable importance in quantum mechanics. The function space of interest in quantum mechanics is the one in which the functions are square integrable in the sense described below. In this chapter, we study the algebra of operators in the space of square integrable functions.

Space of Square Integrable Functions

Consider the space of complex-valued functions of a real variable x (a ≤ x ≤ b) in which the scalar product between the functions g(x) and f (x), denoted by (g(x), f (x)), is defined by

As a consequence of the definition of scalar product given above,

is finite. A function f (x) for which the integral in the equation above is finite is said to be square integrable. It may be verified that the definition of the scalar product given above satisfies all the axioms of scalar product, namely,

In particular, as a consequence of the axioms above, follows the Schwarz inequality (see (2.8)) which, in the present case, assumes the form

An important consequence of this inequality is the result derived in Ex. 4.1 establishing that any linear combination of square integrable functions is also square integrable, which asserts that the space of square integrable functions forms a vector space.