with the supremum norm, then the map is not surjective. (See space). If is surjective, then is said to be a reflexive Banach space. If then the space is a reflexive Banach space. Examples Dual norm for matrices The defined by is self-dual, i.e., its dual norm is The , a special case of the induced norm when , is defined by the maximum singular values of a matrix, that is, has the nuclear norm as its dual norm, which is defined by for any matrix where denote the singular values. If the Schatten -norm on matrices is dual to the Schatten -norm. Finite-dimensional spaces Let be a norm on The associated dual norm, denoted is defined as (This can be shown to be a norm.) The dual norm can be interpreted as the operator norm of interpreted as a matrix, with the norm on , and the absolute value on : From the definition of dual norm we have the inequality which holds for all and The dual of the dual norm is the original norm: we have for all (This need not hold in infinite-dimensional vector spaces.) The dual of the Euclidean norm is the Euclidean norm, since (This follows from the Cauchy–Schwarz inequality; for nonzero the value of that maximises over is ) The dual of the -norm is the -norm: and the dual of the -norm is the -norm. More generally, Hölder's inequality shows that the dual of the -norm is the -norm, where satisfies that is, As another example, consider the - or spectral norm on . The associated dual norm is which turns out to be the sum of the singular values, where This norm is sometimes called the . Lp and ℓp spaces For -norm (also called -norm) of vector is If satisfy then the and norms are dual to each other and the same is true of the and norms, where is some measure space. In particular the Euclidean norm is self-dual since For , the dual norm is with positive definite. For the -norm is even induced by a canonical inner product meaning that for all vectors This inner product can expressed in terms of the norm by using the polarization identity. On this is the defined by while for the space associated with a measure space which consists of all square-integrable functions, this inner product is The norms of the