Let H be a Hilbert space and (e_i)_(i element I) is an orthonormal basis for H. The set S(H) of all operators T for which sum_(i element I) ( left double bracketing bar T e_i right double bracketing bar )^2<∞ is a self-adjoint ideal of B(H). These operators are called Hilbert-Schmidt operators on H. The algebra S(H) with the Hilbert-Schmidt norm left double bracketing bar T right double bracketing bar _2 = sum_(i element I) ( left double bracketing bar T e_i right double bracketing bar )^2 )^(1/2) is a Banach algebra. It contains operators of finite rank as a dense subset and is contained in the space K(H) of compact operators. For any pair of operators T and S in S(H), the family (〈T e_i, S e_i〉)_(i element I) is summable. Its sum (A, B) defines an inner product in S(H) and (T, T)^(1/2) = left double bracketing bar T right double bracketing bar _2. So S(H) can be regarded as a Hilbert space (independent on the choice basis (e_i)).

Hilbert-Schmidt norm | Hilbert space

David Hilbert