Bounded set (topological vector space)
Encyclopedia
In functional analysis
and related areas of mathematics
, a set in a topological vector space
is called bounded or von Neumann bounded, if every neighborhood
of the zero vector can be inflated to include the set. Conversely a set which is not bounded is called unbounded.
Bounded sets are a natural way to define a locally convex polar topologies on the vector space
s in a dual pair
, as the polar of a bounded set is an absolutely convex and absorbing set
. The concept was first introduced by John von Neumann
and Andrey Kolmogorov
in 1935.
(X,τ) over a field
F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar
α so that
with
.
In other words a set is called bounded if it is absorbed
by every neighborhood
of the zero vector.
In locally convex topological vector space
s the topology τ of the space can be specified by a family P of semi-norms. An equivalent characterization of bounded sets in this case is, a set S in (X,P) is bounded if and only if it is bounded for all semi normed spaces (X,p) with p a semi norm of P.
s. A subset A of a topological module M over a topological ring
R is bounded if for any neighborhood N of 0M there exists a neighborhood w of 0R such that w A ⊂ N.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
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 set in a topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
is called bounded or von Neumann bounded, if every neighborhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of the zero vector can be inflated to include the set. Conversely a set which is not bounded is called unbounded.
Bounded sets are a natural way to define a locally convex polar topologies on the vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s in a dual pair
Dual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
, as the polar of a bounded set is an absolutely convex and absorbing set
Absorbing set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space...
. The concept was first introduced by John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
and Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
in 1935.
Definition
Given a topological vector spaceTopological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
(X,τ) over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
α so that
with
.In other words a set is called bounded if it is absorbed
Absorbing set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space...
by every neighborhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of the zero vector.
In locally convex topological vector space
Locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces which generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of ...
s the topology τ of the space can be specified by a family P of semi-norms. An equivalent characterization of bounded sets in this case is, a set S in (X,P) is bounded if and only if it is bounded for all semi normed spaces (X,p) with p a semi norm of P.
Examples and nonexamples
- Every finite set of points is bounded
- The set of points of a Cauchy sequenceCauchy sequenceIn mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
is bounded, the set of points of a Cauchy netNet (mathematics)In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
need not to be bounded. - Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topologyWeak topology (polar topology)In functional analysis and related areas of mathematics the weak topology is the coarsest polar topology, the topology with the fewest open sets, on a dual pair...
the converse is also true. - A (non null) subspace of a Hausdorff topological vector space is not bounded
Properties
- The closureClosure (topology)In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of a bounded set is bounded. - In a locally convex space, the convex envelope of a bounded set is bounded. (Without local convexity this is false, as the
spaces for 
have no nontrivial open convex subsets.) - The finite unionUnion (set theory)In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
or finite sum of bounded sets is bounded. - Continuous linear mappings between topological vector spaces preserve boundedness.
- A locally convex space is seminormable if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there exists a bounded neighbourhood of zero. - The polar of a bounded set is an absolutely convex and absorbing set.
- A set A is bounded if and only if every countable subset of A is bounded
Generalization
The definition of bounded sets can be generalized to topological moduleTopological module
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.- Examples :A topological vector space is a topological module over a topological field....
s. A subset A of a topological module M over a topological ring
Topological ring
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...
R is bounded if for any neighborhood N of 0M there exists a neighborhood w of 0R such that w A ⊂ N.