A rigged Hilbert space is a formal device which makes it possible to treat spectral theory of normal unbounded operators on a Hilbert space as if it were about actual eigenvalues and eigenvectors. It may be used to make rigorous the idea of eigenstates for observables as they commonly arise in quantum mechanics.
Let be the Hilbert space consisting of square integrable functions with respect to Lebesgue measure. There is an unbounded self-adjoint operator
where . This operator is not defined on all of , but it is defined on a dense subspace of . For example, if is the Schwartz space consisting of smooth functions on all of whose derivatives decay rapidly at infinity (more rapidly than any negative power of ), then there is a dense inclusion map , and is defined globally on .
Meanwhile the Schwartz space carries its own topology (as described in the article distribution), stronger than the topology it inherits from , and the space of tempered distributions is defined to be the continuous dual of the TVS . Since the continuous inclusion is dense, it follows that any continuous functional
has at most one extension to a continuous functional . In other words, the adjoint map
is injective. In addition, the topology on is such that the operator is continuous.
In this example, there is a dense inclusion defined by the inner product pairing, and the operator extends uniquely to an operator , called by abuse of notation. Again, in this example, the operator has an eigenvector for each :
Ugh. Lousy start on something that would be nice to understand properly. Maybe an expert can help out. John, you there?
Among the original treatises on the theory of rigged Hilbert spaces is
A unification of various inequivalent approaches is claimed to be achieved in