Orthogonality Spaces Associated with Posets

Function Spaces of Posets with Projections

Inner Product Spaces and Orthogonality

1 Dot product of R The inner product or dot product of R is a function 〈 , 〉 defined by 〈u,v〉 = a1b1 + a2b2 + · · ·+ anbn for u = [a1, a2, . . . , an] , v = [b1, b2, . . . , bn] ∈ R. The inner product 〈 , 〉 satisfies the following properties: (1) Linearity: 〈au + bv,w〉 = a〈u,w〉+ b〈v,w〉. (2) Symmetric Property: 〈u,v〉 = 〈v,u〉. (3) Positive Definite Property: For any u ∈ V , 〈u,u〉 ≥ 0; and 〈u,u〉 =...

Numerical Range and Orthogonality in Normed Spaces

Introducing the concept of the normalized duality mapping on normed linear space and normed algebra, we extend the usual definitions of the numerical range from one operator to two operators. In this note we study the convexity of these types of numerical ranges in normed algebras and linear spaces. We establish some Birkhoff-James orthogonality results in terms of the algebra numerical range V...

Orthogonality and Disjointness in Spaces of Measures

The convex and metric structures underlying probabilistic physical theories are generally described in terms of base normed vector spaces. According to a recent proposal, the purely geometrical features of these spaces are appropriately represented in terms of the notion of measure cone and the mixing distance [1], a specification of the novel concept of direction distance [2]. It turns out tha...

Sign Types Associated to Posets

We start with a combinatorial definition of I-sign types which are a generalization of the sign types indexed by the root system of type Al (I ⊂ N finite). Then we study the set DI p of I-sign types associated to the partial orders on I. We establish a 1-1 correspondence between D [n] p and a certain set of convex simplexes in a euclidean space by which we get a geometric distinction of the sig...