Bounded set (topological vector space)

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 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.
Definition: Given a topological vector space (TVS) formula_4 over a field formula_5 a subset formula_6 of formula_7 is called von Neumann bounded or just bounded in formula_7 if any of the following equivalent conditions is satisfied:
while if formula_7 is a locally convex space whose topology is defined by a family formula_10 of continuous seminorms, then we may add to this list:
while if formula_7 is a seminormed space with seminorm formula_12 (note that every normed space is a seminormed space and every norm is a seminorm), then we may add to this list:
while if formula_6 is a vector subspace of the TVS formula_7 then we may add to this list:
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 formula_7 is called the von Neumann bornology or the (canonical) bornology of formula_7.
A base or fundamental system of bounded sets of formula_7 is a set formula_18 of bounded subsets of formula_7 such that every bounded subset of formula_7 is a subset of some formula_21 
The set of all bounded subsets of formula_7 trivially forms a fundamental system of bounded sets of formula_23
Examples.
In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.
Stability properties.
Let formula_7 be any topological vector space (TVS) (not necessarily Hausdorff or locally convex).
Generalization.
The definition of bounded sets can be generalized to topological modules. 
A subset formula_1 of a topological module formula_26 over a topological ring formula_27 is bounded if for any neighborhood formula_28 of formula_29 there exists a neighborhood formula_30 of formula_31 such that formula_32 