Quotient space
In topology and functional analysis, a quotient space is (intuitively speaking) the result of identifying or "gluing together" certain points of some other space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. The operation can be thought of (very informally indeed) as the act of "dividing" the input space by the equivalence relation, hence both the name "quotient", and the notation, which are both reminiscent of division.
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2 Quotient of a vector space by a subspace 3 Quotient of a Banach space by a subspace |
Quotient of a topological space by an equivalence relation
Formally, suppose X is a topological space and ~ is an equivalence relation on X. We define a topology on the quotient set X/~ (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open if and only if their union is open in X.
Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y iff x-y is an integer. Then the quotient space X/~ (also written as R/Z) is homeomorphic to the unit circle S1.
As another example, consider the unit square X = [0,1]×[0,1] and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then X/~ is homeomorphic to the unit sphere S2.
Let p : X → X/~ be the projection map which sends each element of X to its equivalence class. The map p is continuous; in fact, the topology on X/~ is the finest (the one with the most open sets) which makes p continuous. The map p is in general not open.
If Y is some other topological space, then a function f : X/~ → Y is continuous if and only if fop is continuous.
If g : X → Y is a continuous map with the property that a~b implies g(a)=g(b), then there exists a unique continuous map h : X/~ → Y such that g = hop.
The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is constantly being used when studying quotient spaces.
If X is a vector space, then the quotient space can sometimes also be seen as a vector space.
Specifically, let X be a vector space over K, where either K = R or K = C, and let M be a subspace of X. We define an equivalence relation ~ on X by stating that x ~ y if x − y ∈ M. Let [x] denote the equivalence class containing x. We define scalar multiplication and addition on the equivalent classes by α [x] = [αx] and [x] + [y] = [x+y]. This definition makes sense, because the equivalence classes on the right-hand side do not depend on the elements chosen to represent the equivalence classes on the left-hand side. This turns the quotient space X/M into a vector space.
If '\'X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on X/M'' by
Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1]. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R.
If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
See also: Quotient groupExamples
Properties
Compatibility with other topological notions
please add more results like thisQuotient of a vector space by a subspace
Quotient of a Banach space by a subspace
The quotient space X/M is complete with respect to the norm, so it is a Banach space.Examples