In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field
Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex).
is a convex set then this list may be extended to include: If
Normed and topological vector spaces The open and closed balls centered at the origin in a normed vector space are balanced sets.
is any balanced neighborhood of the origin in a topological vector space
itself, the empty set and the open and closed discs centered at zero.
Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at the origin will do.
are entirely different as far as scalar multiplication is concerned.
is a closed, symmetric, and balanced neighborhood of the origin in
is a closed, symmetric, and balanced neighborhood of the origin in
is a balanced and absorbing set but it is not necessarily convex.
which is a horizontal closed line segment lying above the
is a non-convex subset that is "hour glass shaped" and equal to the union of two closed and filled isosceles triangles
is the filled triangle whose vertices are the origin together with the endpoints of
The Cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field
but whose (nonempty) topological interior does not contain the origin and is therefore not a balanced set.
[6] Similarly for real vector spaces, if
is an (hour glass shaped) balanced subset of
In fact, the following construction produces such balanced sets.
will be convex (respectively, closed, balanced, bounded, a neighborhood of the origin, an absorbing subset of
is a star shaped at the origin,[note 2] which is true, for instance, when
will be a balanced convex neighborhood of the origin and so its topological interior will be a balanced convex open neighborhood of the origin.
is star shaped at the origin[note 2] then so is every
there exists some basic open neighborhood
into a seminormed space that carries its canonical pseduometrizable topology.
as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology.
is a topological vector space and if this convex absorbing subset
does not contain any non-trivial vector subspace then
[8] If a set is closed (respectively, convex, absorbing, a neighborhood of the origin) then the same is true of its balanced core.
on a real or complex vector space is said to be a balanced function if it satisfies any of the following equivalent conditions:[9] If