Hilbert–Schmidt operator
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In mathematics, a Hilbert–Schmidt operator is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm, meaning that there exists an orthonormal basis 
of H with the property
If this is true for one orthonormal basis, it is true for any other orthonormal basis.
Let A and B be two Hilbert–Schmidt operators. The Hilbert–Schmidt inner product can be defined as
The induced norm is called the Hilbert–Schmidt norm:
This definition is independent of the choice of orthonormal basis, and is analogous to the Frobenius norm for operators on a finite-dimensional vector space.
The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. The Hilbert–Schmidt operators are closed in the norm topology if, and only if, H is finite dimensional. They also form a Hilbert space, and can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces
where H* is the dual space of H.
