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A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, [5] are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols.
A subset of a vector space is called a cone if for all real >,.A cone is called pointed if it contains the origin. A cone is convex if and only if +. The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones).
The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff. The left order topology is the standard topology used for many set-theoretic purposes on a Boolean algebra. [clarification needed]
In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, The sample space is typically the set {,,,,,} where each element in the set is a label which represents the outcome of the die landing on that label.
An order unit is an element of an ordered vector space which can be used to bound all elements from above. [1] In this way (as seen in the first example below) the order unit generalizes the unit element in the reals. According to H. H. Schaefer, "most of the ordered vector spaces occurring in analysis do not have order units." [2]
An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. [1] A pre ordered vector space X {\displaystyle X} is called almost Archimedean if for all x ∈ X , {\displaystyle x\in X,} whenever there exists a y ∈ X {\displaystyle y\in X} such that − n − 1 y ≤ x ≤ n − 1 y {\displaystyle -n^{-1 ...
The space of continuous real valued functions with compact support on a topological space with the pointwise partial order defined by when () for all , is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless X {\displaystyle X} satisfies further conditions (for example, being extremally ...
Banach lattices are extremely common in functional analysis, and "every known example [in 1948] of a Banach space [was] also a vector lattice." [1] In particular: ℝ, together with its absolute value as a norm, is a Banach lattice.