Dual norm

In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Let be a normed vector space with norm and let denote its continuous dual space. The dual norm of a continuous linear functional belonging to is the non-negative real number defined by any of the following equivalent formulas: where and denote the supremum and infimum, respectively. The constant map is the origin of the vector space and it always has norm If then the only linear functional on is the constant map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal instead of the correct value of Importantly, a linear function is not, in general, guaranteed to achieve its norm on the closed unit ball meaning that there might not exist any vector of norm such that (if such a vector does exist and if then would necessarily have unit norm ). R.C. James proved James's theorem in 1964, which states that a Banach space is reflexive if and only if every bounded linear function achieves its norm on the closed unit ball. It follows, in particular, that every non-reflexive Banach space has some bounded linear functional that does not achieve its norm on the closed unit ball. However, the Bishop–Phelps theorem guarantees that the set of bounded linear functionals that achieve their norm on the unit sphere of a Banach space is a norm-dense subset of the continuous dual space. The map defines a norm on (See Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. Since the ground field of ( or ) is complete, is a Banach space. The topology on induced by turns out to be stronger than the weak-* topology on The double dual (or second dual) of is the dual of the normed vector space . There is a natural map . Indeed, for each in define The map is linear, injective, and distance preserving. In particular, if is complete (i.e. a Banach space), then is an isometry onto a closed subspace of .