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 matric ...
and related areas of
mathematics 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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
(over a
field with an
absolute value 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 subset
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
because every
neighborhood 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 al ...
(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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
). This neighborhood can also be chosen to be an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
or, alternatively, a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric spac ...
.
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
If
is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
then this list may be extended to include:
- for all scalars satisfying
If
then this list may be extended to include:
- is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
(meaning ) and 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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
. 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 vectors spaces
The open and closed
balls centered at the origin in a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "leng ...
are balanced sets. If
is a
seminorm (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 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 al ...
then
Balanced sets in
and
Let
be the field
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 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
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
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
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
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 or absorbing. However,
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
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
of a closed set need not be closed. Take for instance the graph of
in
The next example shows that the
balanced hull
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
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 triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s
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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ ...
of a family of balanced sets is balanced in the
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-see ...
of the corresponding vector spaces (over the same field
).
- The balanced hull of a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
(respectively, totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “si ...
, bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
) 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
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of balanced sets.
- Scalar multiples and (finite)
Minkowski sum
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set
: A + B = \.
Analogously, the Minkowsk ...
s 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 pr ...
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 al ...
, the closure of a balanced set is balanced. The union of
and the
topological interior of a balanced set is balanced. Therefore, the topological interior of a balanced
neighborhood 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 the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
, 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.
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 al ...
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
In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements.
Definition
In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs ...
will be convex (respectively, closed, balanced,
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
, 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
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...
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, then ...
of
will be a
seminorm on
thereby making
into a
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
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
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
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 al ...
and if this convex absorbing subset
is also a
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of
then the same will be true of the absorbing disk
in which case
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 (pronounced ) 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 ve ...
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
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and balanced.
Every balanced set is
star-shaped
In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defini ...
(at 0) and a
symmetric set
In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements.
Definition
In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs ...
.
If
is a balanced subset of
then
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
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdo ...
; 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
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field \mathbb with an absolute value function , \cdot , ) is a set S such that a S \subseteq S for all scalars a satisfying , a, ...
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 al ...
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 if and only if it is a balanced
sublinear function.
See also
*
*
*
*
*
*
*
References
Proofs
Sources
*
*
*
*
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{{TopologicalVectorSpaces
Linear algebra