Relatively compact subspace
Encyclopedia
In mathematics
, a relatively compact subspace (or relatively compact subset) Y of a topological space
X is a subset whose closure is compact
.
Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In the case of a metric topology, or more generally when sequence
s may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. Such a subset may also be called relatively bounded, or pre-compact, although the latter term is also used for a totally bounded subset. (These are equivalent in a complete space
.)
Some major theorems characterise relatively compact subsets, in particular in function space
s. An example is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family
in complex analysis
. Mahler's compactness theorem
in the geometry of numbers
characterises relatively compact subsets in certain non-compact homogeneous space
s (specifically spaces of lattice
s).
The definition of an almost periodic function
F at a conceptual level has to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a relatively compact subspace (or relatively compact subset) Y of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is a subset whose closure is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
.
Since closed subsets of a compact space are compact, every subset of a compact space is relatively compact. In the case of a metric topology, or more generally when sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. Such a subset may also be called relatively bounded, or pre-compact, although the latter term is also used for a totally bounded subset. (These are equivalent in a complete space
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
.)
Some major theorems characterise relatively compact subsets, in particular in function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
s. An example is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family
Normal family
In mathematics, with special application to complex analysis, a normal family is a pre-compact family of continuous functions. Informally, this means that the functions in the family are not exceedingly numerous or widely spread out; rather, they stick together in a relatively "compact" manner...
in complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
. Mahler's compactness theorem
Mahler's compactness theorem
In mathematics, Mahler's compactness theorem, proved by , is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in which a lattice could degenerate in a sequence of...
in the geometry of numbers
Geometry of numbers
In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by ....
characterises relatively compact subsets in certain non-compact homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
s (specifically spaces of lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
s).
The definition of an almost periodic function
Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov,...
F at a conceptual level has to do with the translates of F being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.