Tensor Products of Hilbert Spaces
The algebraic tensor product of two Hilbert spaces A and B has a natural positive definite sesquilinear form induced by the sesquilinear forms of A and B. So in particular it has a natural positive definite quadratic form, and the corresponding completion is a Hilbert space A⊗B, called the (Hilbert space) tensor product of A and B.
If the vectors ai and bj run through orthonormal bases of A and B, then the vectors ai⊗bj form an orthonormal basis of A⊗B.
Read more about this topic: Topological Tensor Product
Famous quotes containing the words products and/or spaces:
“All that is told of the sea has a fabulous sound to an inhabitant of the land, and all its products have a certain fabulous quality, as if they belonged to another planet, from seaweed to a sailor’s yarn, or a fish story. In this element the animal and vegetable kingdoms meet and are strangely mingled.”
—Henry David Thoreau (1817–1862)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)