operators is closed in the norm topology if, and only if, is finite-dimensional. Properties Every Hilbert–Schmidt operator is a compact operator. A bounded linear operator is Hilbert–Schmidt if and only if the same is true of the operator , in which case the Hilbert–Schmidt norms of T and |T| are equal. Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact operators. If and are Hilbert–Schmidt operators between Hilbert spaces then the composition is a nuclear operator. If is a bounded linear operator then we have . is a Hilbert–Schmidt operator if and only if the trace of the nonnegative self-adjoint operator is finite, in which case . If is a bounded linear operator on and is a Hilbert–Schmidt operator on then , , and . In particular, the composition of two Hilbert–Schmidt operators is again Hilbert–Schmidt (and even a trace class operator). The space of Hilbert–Schmidt operators on is an ideal of the space of bounded operators that contains the operators of finite-rank. If is a Hilbert–Schmidt operator on then where is an orthonormal basis of H, and is the Schatten norm of for . In Euclidean space, is also called the Frobenius norm. See also References Linear operators Operator theory