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Concept# Bounded set (topological vector space)

Summary

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.
A set that is not bounded is called unbounded.
Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set 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.
Suppose is a topological vector space (TVS) over a field
A subset of is called or just in if any of the following equivalent conditions are satisfied:
For every neighborhood of the origin there exists a real such that for all scalars satisfying
This was the definition introduced by John von Neumann in 1935.
is absorbed by every neighborhood of the origin.
For every neighborhood of the origin there exists a scalar such that
For every neighborhood of the origin there exists a real such that for all scalars satisfying
For every neighborhood of the origin there exists a real such that for all real
Any one of statements (1) through (5) above but with the word "neighborhood" replaced by any of the following: "balanced neighborhood," "open balanced neighborhood," "closed balanced neighborhood," "open neighborhood," "closed neighborhood".
e.g. Statement (2) may become: is bounded if and only if is absorbed by every balanced neighborhood of the origin.
If is locally convex then the adjective "convex" may be also be added to any of these 5 replacements.
For every sequence of scalars that converges to and every sequence in the sequence converges to in
This was the definition of "bounded" that Andrey Kolmogorov used in 1934, which is the same as the definition introduced by Stanisław Mazur and Władysław Orlicz in 1933 for metrizable TVS. Kolmogorov used this definition to prove that a TVS is seminormable if and only if it has a bounded convex neighborhood of the origin.

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