Published online by Cambridge University Press: 24 October 2008
Let denote a Hilbert space (real or complex), with inner product (|). In order to present our notation, we recall that if
is a vector subspace of
(and ‘vector sub-spaces’ will always be closed), with orthocomplement
, a partial isometry V with initial domain
is a linear operator in
which preserves length, and so inner-product, in
and is zero in
is the final domain of V, and it is easy to verify that V*, the adjoint operator, is also a partial isometry, with initial domain
and final domain
.