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In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe" [1]) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair (,) is called a measurable space.
Apart from the trivial case of a zero-dimensional space over any field, a vector space over a field F has a finite number of elements if and only if F is a finite field and the vector space has a finite dimension. Thus we have F q, the unique finite field (up to isomorphism) with q elements. Here q must be a power of a prime (q = p m with p prime).
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function.
The simplest example of a vector space over a field F is the field F itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all n -tuples (sequences of length n ) ( a 1 , a 2 , … , a n ) {\displaystyle (a_{1},a_{2},\dots ,a_{n})} of elements a i of F form a vector space that ...
(More generally, any field of sets forms a group with the symmetric difference as operation.) A group in which every element is its own inverse (or, equivalently, in which every element has order 2) is sometimes called a Boolean group ; [ 2 ] [ 3 ] the symmetric difference provides a prototypical example of such groups.
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In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
Let F be a field and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define (+) = + () = When the domain X has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure.