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.
The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
is a topological vector space (TVS) over a field
at the origin then this list may be extended to include: If
is a locally convex space whose topology is defined by a family
of continuous seminorms, then this list may be extended to include: If
is merely a seminorm),[note 2] then this list may be extended to include: If
then this list may be extended to include: A subset that is not bounded is called unbounded.
The collection of all bounded sets on a topological vector space
A base or fundamental system of bounded sets of
trivially forms a fundamental system of bounded sets of
In any locally convex TVS, the set of closed and bounded disks are a base of bounded set.
[1] Unless indicated otherwise, a topological vector space (TVS) need not be Hausdorff nor locally convex.
Any vector subspace of a TVS that is not a contained in the closure of
is unbounded There exists a Fréchet space
[6] A locally convex topological vector space has a bounded neighborhood of zero if and only if its topology can be defined by a single seminorm.
is a countable sequence of bounded subsets of a metrizable locally convex topological vector space
Using the definition of uniformly bounded sets given below, Mackey's countability condition can be restated as: If
are bounded subsets of a metrizable locally convex space then there exists a sequence
In words, given any countable family of bounded sets in a metrizable locally convex space, it is possible to scale each set by its own positive real so that they become uniformly bounded.
In the case of a normed (or seminormed) space, a family
is norm bounded, meaning that there exists some real
which by definition means that there exists some bounded subset
of linear maps between two normed (or seminormed) spaces
be a set of continuous linear operators between two topological vector spaces
being bounded guarantees the existence of some positive integer
is an equicontinuous set of continuous linear operators between two topological vector spaces
(not necessarily Hausdorff or locally convex), then the orbit
The definition of bounded sets can be generalized to topological modules.