Lanczos iterations). The norm of the adjoint or transpose can be computed as follows. We have that for any then where are Hölder conjugate to that is, and Operators on a Hilbert space Suppose is a real or complex Hilbert space. If is a bounded linear operator, then we have and where denotes the adjoint operator of (which in Euclidean spaces with the standard inner product corresponds to the conjugate transpose of the matrix ). In general, the spectral radius of is bounded above by the operator norm of : To see why equality may not always hold, consider the Jordan canonical form of a matrix in the finite-dimensional case. Because there are non-zero entries on the superdiagonal, equality may be violated. The quasinilpotent operators is one class of such examples. A nonzero quasinilpotent operator has spectrum So while However, when a matrix is normal, its Jordan canonical form is diagonal (up to unitary equivalence); this is the spectral theorem. In that case it is easy to see that This formula can sometimes be used to compute the operator norm of a given bounded operator : define the Hermitian operator determine its spectral radius, and take the square root to obtain the operator norm of The space of bounded operators on with the topology induced by operator norm, is not separable. For example, consider the Lp space which is a Hilbert space. For let be the characteristic function of and be the multiplication operator given by that is, Then each is a bounded operator with operator norm 1 and But is an uncountable set. This implies the space of bounded operators on is not separable, in operator norm. One can compare this with the fact that the sequence space is not separable. The associative algebra of all bounded operators on a Hilbert space, together with the operator norm and the adjoint operation, yields a C*-algebra. See also Notes References . Functional analysis Norms (mathematics) Operator theory