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Every norm is a restriction of an order-unit norm

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We point out the equivalence of the fact that every norm on a vector space is a restriction of an order-unit norm to that of Paulsen’s construction concerning generalization of operator systems. | Turk J Math (2016) 40: 1398 – 1400 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1510-55 Research Article Every norm is a restriction of an order-unit norm 1 1 ˘ Mert C ¸ AGLAR , Zafer ERCAN2,∗ ˙ ˙ Department of Mathematics and Computer Science, Istanbul K¨ ult¨ ur University, Bakırk¨ oy, Istanbul, Turkey 2 ˙ Department of Mathematics, Abant Izzet Baysal University, Bolu, Turkey Received: 15.10.2015 • Accepted/Published Online: 26.02.2016 • Final Version: 02.12.2016 Abstract: We point out the equivalence of the fact that every norm on a vector space is a restriction of an order-unit norm to that of Paulsen’s construction concerning generalization of operator systems. Key words: Norm, ordered vector space, order-unit 1. Introduction The purpose of this very short expository note is to bring a widely unnoticed fact concerning normed spaces to the readers’ attention by pointing out that it is equivalent to Paulsen’s construction in quantum analysis given in [4]. We refer to [1] for the general theory of ordered vector spaces. A subset K of a vector space E is called a cone if K + K ⊆ K, R+ K ⊆ K, and K ∩ (−K) = {0}, in which case the pair (E, K) is called an ordered vector space. We write x ⩽ y , or y ⩾ x in E , whenever y − x ∈ K . An element e ∈ K ∖ {0} is called an order-unit if for each x ∈ E there exists a λ > 0 such that x ⩽ λe . The notion of order-unit is due to Kadison [3]. An ordered vector space E is called almost Archimedean if −εx ⩽ y ⩽ εx for all ε > 0 ; then y = 0 . Similarly, E is called Archimedean if Nx ⩽ y implies x ⩽ 0. It is obvious that Archimedeanness implies almost Archimedeanness, but not vice versa. If (E, K) is an almost Archimedean vector space with an order-unit e > 0, then ∥x∥e = inf{ε > 0 : −εe ⩽ x ⩽ εe} defines a norm on the ordered vector space (E, K). Let us call this norm as the norm generated by the order unit e . Theorem 1 Let (E, ∥ · ∥) be a normed space. Then .