Given an operator on Hilbert space , consider the orbit of a point in under the iterates of . Applying Gram–Schmidt one obtains an orthonormal basis for . Let be the corresponding nested sequence of "coordinate" subspaces of . The matrix expressing with respect to is almost upper triangular, in the sense that the coefficients are the only nonzero sub-diagonal coefficients. Bernstein and Robinson show that if is polynomially compact, then there is a hyperfinite index such that the matrix coefficient is infinitesimal. Next, consider the subspace of . If in has finite norm, then is infinitely close to .