In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
and related areas of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
a balanced set, circled set or disk in a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
(over a
field with an
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
function
) is a
set such that
for all
scalars
satisfying
The balanced hull or balanced envelope of a set
is the smallest balanced set containing
The balanced core of a set
is the largest balanced set contained in
Balanced sets are ubiquitous in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
because every
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of the origin in every
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(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). This neighborhood can also be chosen to be an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
or, alternatively, a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
.
Definition
Let
be a vector space over the
field of
real or
complex numbers.
Notation
If
is a set,
is a scalar, and
then let
and
and for any
let
denote, respectively, the ''open ball'' and the ''closed ball'' of radius
in the scalar field
centered at
where
and
Every balanced subset of the field
is of the form
or
for some
Balanced set
A subset
of
is called a ' or ''balanced'' if it satisfies any of the following equivalent conditions:
- ''Definition'': for all and all scalars satisfying
- for all scalars satisfying
- (where ).
- For every
* is a (if ) or (if ) dimensional vector subspace of
* If then the above equality becomes which is exactly the previous condition for a set to be balanced. Thus, is balanced if and only if for every is a balanced set (according to any of the previous defining conditions).
- For every 1-dimensional vector subspace of is a balanced set (according to any defining condition other than this one).
- For every there exists some such that or
- is a balanced subset of (according to any defining condition of "balanced" other than this one).
* Thus is a balanced subset of if and only if it is balanced subset of every (equivalently, of some) vector space over the field that contains So assuming that the field is clear from context, this justifies writing " is balanced" without mentioning any vector space.
[Assuming that all vector spaces containing a set are over the same field, when describing the set as being "balanced", it is not necessary to mention a vector space containing That is, " is balanced" may be written in place of " is a balanced subset of ".]
If
is a
convex set then this list may be extended to include:
- for all scalars satisfying
If
then this list may be extended to include:
- is symmetric (meaning ) and
Balanced hull
The ' of a subset
of
denoted by
is defined in any of the following equivalent ways:
- ''Definition'': is the smallest (with respect to ) balanced subset of containing
- is the Intersection (set theory)">intersection of all balanced sets containing
Balanced core
The ' of a subset
of
denoted by
is defined in any of the following equivalent ways:
- ''Definition'': is the largest (with respect to ) balanced subset of
- is the union of all balanced subsets of
- if while if
Examples
The empty set is a balanced set. As is any vector subspace of any (real or complex)
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
. In particular,
is always a balanced set.
Any non-empty set that does not contain the origin is not balanced and furthermore, the
balanced core of such a set will equal the empty set.
Normed and topological vector spaces
The open and closed
balls centered at the origin in a
normed vector space are balanced sets. If
is a
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
(or
norm) on a vector space
then for any constant
the set
is balanced.
If
is any subset and
then
is a balanced set.
In particular, if
is any balanced
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of the origin in a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
then
Balanced sets in
and
Let
be the field
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
let
denote the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
on
and let
denotes the vector space over
So for example, if
is the field of complex numbers then
is a 1-dimensional complex vector space whereas if
then
is a 1-dimensional real vector space.
The balanced subsets of
are exactly the following:
- for some real
- for some real
Consequently, both the
balanced core and the
balanced hull of every set of scalars is equal to one of the sets listed above.
The balanced sets are
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. As a result,
and
are entirely different as far as
scalar multiplication is concerned.
Balanced sets in
Throughout, let
(so
is a vector space over
) and let
is the closed unit ball in
centered at the origin.
If
is non-zero, and
then the set
is a closed, symmetric, and balanced neighborhood of the origin in
More generally, if
is closed subset of
such that
then
is a closed, symmetric, and balanced neighborhood of the origin in
This example can be generalized to
for any integer
Let
be the union of the line segment between the points
and
and the line segment between
and
Then
is balanced but not convex. Nor is
is absorbing (despite the fact that
is the entire vector space).
For every
let
be any positive real number and let
be the (open or closed) line segment in
between the points
and
Then the set
is a balanced and absorbing set but it is not necessarily convex.
The
balanced hull of a closed set need not be closed. Take for instance the graph of
in
The next example shows that the
balanced hull of a convex set may fail to be convex (however, the convex hull of a balanced set is always balanced). For an example, let the convex subset be
which is a horizontal closed line segment lying above the
axis in
The balanced hull
is a non-convex subset that is "
hour glass shaped" and equal to the union of two closed and filled
isosceles triangles
and
where
and
is the filled triangle whose vertices are the origin together with the endpoints of
(said differently,
is the
convex hull of
while
is the convex hull of
).
Sufficient conditions
A set
is balanced if and only if it is equal to its balanced hull
or to its balanced core
in which case all three of these sets are equal:
The
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of a family of balanced sets is balanced in the
product space of the corresponding vector spaces (over the same field
).
- The balanced hull of a compact (respectively, totally bounded, bounded) set has the same property.
- The convex hull of a balanced set is convex and balanced (that is, it is absolutely convex). However, the balanced hull of a convex set may fail to be convex (a counter-example is given above).
- Arbitrary unions of balanced sets are balanced, and the same is true of arbitrary intersections of balanced sets.
- Scalar multiples and (finite) Minkowski sums of balanced sets are again balanced.
- Images and preimages of balanced sets under
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...
s are again balanced. Explicitly, if is a linear map and and are balanced sets, then and are balanced sets.
Balanced neighborhoods
In any
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
, the closure of a balanced set is balanced. The union of the origin
and the
topological interior of a balanced set is balanced. Therefore, the topological interior of a balanced
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of the origin is balanced.
[Let be balanced. If its topological interior is empty then it is balanced so assume otherwise and let be a scalar. If then the map defined by is a ]homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
, which implies because is open, so that it only remains to show that this is true for However, might not be true but when it is true then will be balanced. However,
is a balanced subset of
that contains the origin
but whose (nonempty) topological interior does not contain the origin and is therefore not a balanced set. Similarly for real vector spaces, if
denotes the convex hull of
and
(a filled
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
whose vertices are these three points) then
is an (
hour glass shaped) balanced subset of
whose non-empty topological interior does not contain the origin and so is not a balanced set (and although the set
formed by adding the origin is balanced, it is neither an open set nor a neighborhood of the origin).
Every neighborhood (respectively, convex neighborhood) of the origin in a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
contains a balanced (respectively, convex and balanced) open neighborhood of the origin. In fact, the following construction produces such balanced sets. Given
the
symmetric set will be convex (respectively, closed, balanced,
bounded, a neighborhood of the origin, an
absorbing subset of
) whenever this is true of
It will be a balanced set if
is a
star shaped at the origin,
[ being star shaped at the origin means that and for all and ] which is true, for instance, when
is convex and contains
In particular, if
is a convex neighborhood of the origin then
will be a convex neighborhood of the origin and so its
topological interior will be a balanced convex
neighborhood of the origin.
Suppose that
is a convex and
absorbing subset of
Then
will be
convex balanced absorbing subset of
which guarantees that the
Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If K is a subset of a real or complex vector space X, ...
of
will be a
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on
thereby making
into a
seminormed space that carries its canonical
pseduometrizable topology. The set of scalar multiples
as
ranges over
(or over any other set of non-zero scalars having
as a limit point) forms a neighborhood basis of absorbing
disks at the origin for this
locally convex topology. If
is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
and if this convex absorbing subset
is also a
bounded subset of
then the same will be true of the absorbing disk
if in addition
does not contain any non-trivial vector subspace then
will be a
norm and
will form what is known as an
auxiliary normed space. If this normed space is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
then
is called a .
Properties
Properties of balanced sets
A balanced set is not empty if and only if it contains the origin.
By definition, a set is
absolutely convex if and only if it is
convex and balanced.
Every balanced set is
star-shaped (at 0) and a
symmetric set.
If
is a balanced subset of
then:
- for any scalars and if then and Thus if and are any scalars then
- is absorbing in if and only if for all there exists such that
- for any 1-dimensional vector subspace of the set is convex and balanced. If is not empty and if is a 1-dimensional vector subspace of then is either or else it is absorbing in
- for any if contains more than one point then it is a convex and balanced neighborhood of in the 1-dimensional vector space when this space is endowed with the Hausdorff Euclidean topology; and the set is a convex balanced subset of the real vector space that contains the origin.
Properties of balanced hulls and balanced cores
For any collection
of subsets of
In any topological vector space, the
balanced hull of any open neighborhood of the origin is again open.
If
is a
Hausdorff topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
and if
is a compact subset of
then the balanced hull of
is compact.
If a set is closed (respectively, convex,
absorbing, a neighborhood of the origin) then the same is true of its balanced core.
For any subset
and any scalar
For any scalar
This equality holds for
if and only if
Thus if
or
then
for every scalar
Related notions
A function
on a real or complex vector space is said to be a if it satisfies any of the following equivalent conditions:
- whenever is a scalar satisfying and
- whenever and are scalars satisfying and
- is a balanced set for every non-negative real
If
is a balanced function then
for every scalar
and vector
so in particular,
for every unit length scalar
(satisfying
) and every
Using
shows that every balanced function is a symmetric function.
A real-valued function
is a
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
if and only if it is a balanced
sublinear function
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a semino ...
.
See also
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References
Proofs
Sources
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{{TopologicalVectorSpaces
Linear algebra