Vuong's closeness test

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.

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 spaces 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.

Definition

Given a topological vector space (X,τ) over a field F, S is called bounded if for every neighborhood N of the zero vector there exists a scalar α such that

$S\subseteq \alpha N$

with

$\alpha N:=\{\alpha x\mid x\in N\}$.

In other words a set is called bounded if it is absorbed by every neighborhood of the zero vector.

In locally convex topological vector spaces 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 sequence is bounded, the set of points of a Cauchy net need not to be bounded.
• Every relatively compact set in a topological vector space is bounded. If the space is equipped with the weak topology the converse is also true.
• A (non null) subspace of a Hausdorff topological vector space is not bounded

Properties

• The closure 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 $L^p$ spaces for $0<p<1$ have no nontrivial open convex subsets.)
• The finite union 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 if 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 modules. A subset A of a topological module M over a topological ring R is bounded if for any neighborhood N of 0_M there exists a neighborhood w of 0_R such that w A ⊂ N.

See also

• Totally bounded space
• Local boundedness
• bounded function
• bounding point

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534