ABSTRACT
Operators have already been introduced in finite-dimensional spaces. Many of the definitions and properties of operators remain applicable in a Hilbert space. However, the infinite-dimensional nature of a general Hilbert space gives rise to some serious complications. An example is the non-existence of eigenvalues and eigenvectors for some operators. Another is the emergence of unbounded operators. A Hilbert space admits many different dense subsets, and a dense subset itself may also contain many subsets which are also dense. The domain of a bounded operator can be uniquely extended to the entire Hilbert space. So, we shall assume from now on that bounded operators are defined on the entire Hilbert space. Unbounded operators are generally defined on a domain smaller than the entire Hilbert space. This is because for an arbitrary input vector the operator may produce an output vector with an infinite norm.
