continuous dual spaces of and satisfy the polarization identity, and so these dual norms can be used to define inner products. With this inner product, this dual space is also a Hilbert spaces. Properties Given normed vector spaces and let be the collection of all bounded linear mappings (or ) of into Then can be given a canonical norm. A subset of a normed space is bounded if and only if it lies in some multiple of the unit sphere; thus for every if is a scalar, then so that The triangle inequality in shows that for every satisfying This fact together with the definition of implies the triangle inequality: Since is a non-empty set of non-negative real numbers, is a non-negative real number. If then for some which implies that and consequently This shows that is a normed space. Assume now that is complete and we will show that is complete. Let be a Cauchy sequence in so by definition as This fact together with the relation implies that is a Cauchy sequence in for every It follows that for every the limit exists in and so we will denote this (necessarily unique) limit by that is: It can be shown that is linear. If , then for all sufficiently large integers and . It follows that for sufficiently all large Hence so that and This shows that in the norm topology of This establishes the completeness of When is a scalar field (i.e. or ) so that is the dual space of Let denote the closed unit ball of a normed space When is the scalar field then so part (a) is a corollary of Theorem 1. Fix There exists such that but, for every . (b) follows from the above. Since the open unit ball of is dense in , the definition of shows that if and only if for every . The proof for (c) now follows directly. As usual, let denote the canonical metric induced by the norm by the norm on and denote the distance from a point to the subset by If is a bounded linear functional on a normed space then for every vector where denotes the kernel of See also Notes References External links Notes on the proximal mapping by Lieven Vandenberge Functional analysis Linear algebra Mathematical optimization