Definition: Given a topological vector space over a field, a subset of is called von Neumann bounded or just bounded in if any of the following equivalent conditions is 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.
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 ;
Any of the above 4 conditions 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. Condition 2 may become: is bounded if and only if is absorbed by every balanced neighborhood of the origin.
for every sequence of scalars > that converges to 0 and every sequence in, the sequence converges to 0 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 0.
while if is a vector subspace of the TVS then we may add to this list:
is contained in the closure of.
Definition: A subset that is not bounded is called unbounded.
Bornology and fundamental systems of bounded sets
The collection of all bounded sets on a topological vector space is called the von Neumann bornology or the bornology of . A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some. The set of all bounded subsets of trivially forms a fundamental system of bounded sets of.
Examples
In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.
Stability properties
Let be any topological vector space .
In any TVS, finite unions, finite sums, scalar multiples, subsets, closures, interiors, and balanced hulls of bounded sets are again bounded.
In any locally convex TVS, the convex hull of a bounded set is again bounded. This may fail to be true if the space is not locally convex.
In any TVS, every subset of the closure of is bounded.
Non-examples
In any TVS, any vector subspace that is not a contained in the closure of is unbounded.
There exists a Fréchet space having a bounded subset and also a dense vector subspace such that is not contained in the closure of any bounded subset of.
Properties
Finite unions, finite sums, closures, interiors, and balanced hulls of bounded sets are bounded.
The image of a bounded set under a continuous linear map is bounded.
Without local convexity this is false, as the Lp space| spaces for have no nontrivial open convex subsets.
A locally convex space has a bounded neighborhood of zero if and only if its topology can be defined by a single seminorm.
The polar of a bounded set is an absolutely convex and absorbing set.
Generalization
The definition of bounded sets can be generalized to topological modules. A subset of a topological module over a topological ring is bounded if for any neighborhood of there exists a neighborhood of such that.
