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:
“We are the products of editing, rather than of authorship.”
—George Wald (b. 1906)
“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)