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

\mathbb

with an absolute value function

, \cdot ,

) is a set

S

such that

a S \subseteq S

for all scalars

a

satisfying

, a,  \leq 1.

The balanced hull or balanced envelope of a set

S

is the smallest balanced set containing

S.

The balanced core of a subset

S

is the largest balanced set contained in

S.

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

X

be a vector space over the field

\mathbb

of real or complex numbers. Notation If

S

is a set,

a

is a scalar, and

B \subseteq \mathbb

then let

a S = \

and

B S = \

and for any

0 \leq r \leq \infty,

let

B_r = \ \qquad \text \qquad B_ = \.

denote, respectively, the ''open ball'' and the ''closed ball'' of radius

r

in the scalar field

\mathbb

centered at

0

where

B_0 = \varnothing, B_ = \,

and

B_ = B_ = \mathbb.

Every balanced subset of the field

\mathbb

is of the form

B_

B_r

for some

0 \leq r \leq \infty.

Balanced set A subset

S

X

is called a ' or ''balanced'' if it satisfies any of the following equivalent conditions:

''Definition'': $a s \in S$ for all $s \in S$ and all scalars $a$ satisfying $, a, \leq 1.$
$a S \subseteq S$ for all scalars $a$ satisfying $, a, \leq 1.$
$B_ S \subseteq S,$ where $B_ := \.$
$S = B_ S.$
For every $s \in S,$ $S \cap \mathbb s = B_ (S \cap \mathbb s).$ * $\mathbb s = \operatorname \$ is a $0$ (if $s = 0$ ) or $1$ (if $s \neq 0$ ) dimensional vector subspace of $X.$ * If $R := S \cap \mathbb s$ then the above equality becomes $R = B_ R,$ which is exactly the previous condition for a set to be balanced. Thus, $S$ is balanced if and only if for every $s \in S,$ $A \cap \mathbb s$ is a balanced set (according to any of the previous defining conditions).
For every 1-dimensional vector subspace $Y$ of $\operatorname S,$ $S \cap Y$ is a balanced set (according to any defining condition other than this one).
For every $s \in S,$ there exists some $0 \leq r \leq \infty$ such that $S \cap \mathbb s = B_r s$ or $S \cap \mathbb s = B_ s.$

S

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:

$a S \subseteq S$ for all scalars $a$ satisfying $, a, = 1.$

\mathbb = \R

then this list may be extended to include:

$S$ 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 $- S = S$ ) and
intersection 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, thei ...
of all balanced sets containing $S.$
$\operatorname S = \bigcup_ (a S).$
$\operatorname S = B_ S.$

Balanced core

\operatorname S ~=~ \begin
\displaystyle\bigcap_ a S & \text 0 \in S \\
\varnothing              & \text 0 \not\in S \\
\end

The ' of a subset

S

X,

denoted by

\operatorname S,

is defined in any of the following equivalent ways:

''Definition'': $\operatorname S$ is the largest (with respect to $\,\subseteq\,$ ) balanced subset of $S.$
$\operatorname S$ is the union of all balanced subsets of $S.$
$\operatorname S = \varnothing$ if $0 \not\in S$ while $\operatorname S = \bigcap_ (a S)$ if $0 \in S.$

Examples

The empty set is a balanced set. As is any vector subspace of any (real or complex)

. 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

p

is a seminorm (or norm) on a vector space

X

then for any constant

c > 0,

the set

\

is balanced. If

S \subseteq X

is any subset and

B_1 := \

then

B_1 S

is a balanced set. In particular, if

U \subseteq X

is any balanced neighborhood of the origin in a

X

then

\operatorname_X U ~\subseteq~ B_1 U ~=~ \bigcup_ a U ~\subseteq~ U.

Balanced sets in

\R

and

\Complex

Let

\mathbb

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 ...

\R

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 ...

\Complex,

let

, \cdot,

denote the absolute value on

\mathbb,

and let

X := \mathbb

denotes the vector space over

\mathbb.

So for example, if

\mathbb := \Complex

is the field of complex numbers then

X = \mathbb = \Complex

is a 1-dimensional complex vector space whereas if

\mathbb := \R

then

X = \mathbb = \R

is a 1-dimensional real vector space. The balanced subsets of

X = \mathbb

are exactly the following:

$\varnothing$
$X$
$\$
$\$ for some real $r > 0$
$\$ for some real $r > 0.$

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

\Complex

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,

\Complex

and

\R^2

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

\R^2

Throughout, let

X = \R^2

(so

X

is a vector space over

\R

) and let

B_

is the closed unit ball in

X

centered at the origin. If

x_0 \in X = \R^2

is non-zero, and

L := \R x_0,

then the set

R := B_ \cup L

is a closed, symmetric, and balanced neighborhood of the origin in

X.

More generally, if

C

is closed subset of

X

such that

(0, 1) C \subseteq C,

then

S := B_ \cup C \cup (-C)

is a closed, symmetric, and balanced neighborhood of the origin in

X.

This example can be generalized to

\R^n

for any integer

n \geq 1.

Let

B \subseteq \R^2

be the union of the line segment between the points

(-1, 0)

and

(1, 0)

and the line segment between

(0, -1)

and

(0, 1).

Then

B

is balanced but not convex or absorbing. However,

\operatorname B = \R^2.

For every

0 \leq t \leq \pi,

let

r_t

be any positive real number and let

B^t

be the (open or closed) line segment in

X := \R^2

between the points

(\cos t, \sin t)

and

- (\cos t, \sin t).

Then the set

B = \bigcup_ r_t B^t

is a balanced and absorbing set but it is not necessarily convex. The

of a closed set need not be closed. Take for instance the graph of

x y = 1

X = \R^2.

The next example shows that the

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

\times \,

which is a horizontal closed line segment lying above the

x-

axis in

X := \R^2.

The balanced hull

\operatorname S

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 ...

T_1

and

T_2,

where

T_2 = - T_1

and

T_1

is the filled triangle whose vertices are the origin together with the endpoints of

S

(said differently,

T_1

is the convex hull of

S \cup \

while

T_2

is the convex hull of

(-S) \cup \

Sufficient conditions

A set

T

is balanced if and only if it is equal to its balanced hull

\operatorname T

or to its balanced core

\operatorname T,

in which case all three of these sets are equal:

T = \operatorname T = \operatorname T.

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

\mathbb

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 $L : X \to Y$ is a linear map and $B \subseteq X$ and $C \subseteq Y$ are balanced sets, then $L(B)$ and $L^(C)$ are balanced sets.

Balanced neighborhoods

In any

, 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

B \subseteq X

be balanced. If its topological interior

\operatorname_X B

is empty then it is balanced so assume otherwise and let

, s,  \leq 1

be a scalar. If

s \neq 0

then the map

X \to X

defined by

x \mapsto s x

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

s \operatorname_X B = \operatorname_X (s B) \subseteq s B \subseteq B;

because

s \operatorname_X B

is open,

s \operatorname_X B \subseteq \operatorname_X B

so that it only remains to show that this is true for

s = 0.

However,

0 \in \operatorname_X B

might not be true but when it is true then

\operatorname_X B

will be balanced.

\blacksquare

However,

\left\

is a balanced subset of

X = \Complex^2

that contains the origin

(0, 0) \in X

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

X

contains a balanced (respectively, convex and balanced) open neighborhood of the origin. In fact, the following construction produces such balanced sets. Given

W \subseteq X,

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 ...

\bigcap_ u W \subseteq W

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

X

) whenever this is true of

W.

It will be a balanced set if

W

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,

W

being star shaped at the origin means that

0 \in W

and

r w \in W

for all

0 \leq r \leq 1

and

w \in W.

which is true, for instance, when

W

is convex and contains

0.

In particular, if

W

is a convex neighborhood of the origin then

\bigcap_ u W

will be a convex neighborhood of the origin and so its topological interior will be a balanced convex neighborhood of the origin. Suppose that

W

is a convex and absorbing subset of

X.

Then

D := \bigcap_ u W

will be convex balanced absorbing subset of

X,

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 ...

p_D : X \to \R

D

will be a seminorm on

X,

thereby making

\left(X, p_D\right)

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

r D

r

ranges over

\left\

(or over any other set of non-zero scalars having

0

as a limit point) forms a neighborhood basis of absorbing disks at the origin for this

topology. If

X

is a

and if this convex absorbing subset

W

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 ...

X,

then the same will be true of the absorbing disk

D := \bigcap_ u W,

in which case

p_D

will be a norm and

\left(X, p_D\right)

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

D

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

. If

B

is a balanced subset of

X

then If

B

is a balanced subset of

X

then:

for any scalars $c$ and $d,$ if $, c, \leq , d,$ then $c B \subseteq d B$ and $c B = , d, B.$ Thus if $c$ and $d$ are any scalars then $(c B) \cap (d B) = \min_ \ B.$
$B$ is absorbing in $X$ if and only if for all $x \in X,$ there exists $r > 0$ such that $x \in r B.$
for any 1-dimensional vector subspace $Y$ of $X,$ the set $B \cap Y$ is convex and balanced. If $B$ is not empty and if $Y$ is a 1-dimensional vector subspace of $\operatorname B$ then $B \cap Y$ is either $\$ or else it is absorbing in $Y.$
for any $x \in X,$ if $B \cap \operatorname x$ contains more than one point then it is a convex and balanced neighborhood of $0$ in the 1-dimensional vector space $\operatorname x$ 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 $B \cap \R x$ is a convex balanced subset of the real vector space $\R x$ that contains the origin.

Properties of balanced hulls and balanced cores For any collection

\mathcal

of subsets of

X,

\operatorname \left(\bigcup_ S\right) = \bigcup_ \operatorname S 
\quad \text \quad \operatorname \left(\bigcap_ S\right) = \bigcap_ \operatorname S.

In any topological vector space, the

of any open neighborhood of the origin is again open. If

X

is a Hausdorff

and if

K

is a compact subset of

X

then the balanced hull of

K

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

S \subseteq X

and any scalar

c,

\operatorname (c \, S) = c \operatorname S = , c,  \operatorname S.

For any scalar

c \neq 0,

\operatorname (c \, S) = c \operatorname S = , c,  \operatorname S.

This equality holds for

c = 0

if and only if

S \subseteq \.

Thus if

0 \in S

S = \varnothing

then

\operatorname (c \, S) = c \operatorname S = , c,  \operatorname S

for every scalar

c.

Related notions

A function

p : X \to [0, \infty)

on a real or complex vector space is said to be a if it satisfies any of the following equivalent conditions:

$p(a x) \leq p(x)$ whenever $a$ is a scalar satisfying $, a, \leq 1$ and $x \in X.$
$p(a x) \leq p(b x)$ whenever $a$ and $b$ are scalars satisfying $, a, \leq , b,$ and $x \in X.$
$\$ is a balanced set for every non-negative real $t \geq 0.$

p

is a balanced function then

p(a x) = p(, a,  x)

for every scalar

a

and vector

x \in X;

so in particular,

p(u x) = p(x)

for every unit length scalar

u

(satisfying

, u,  = 1

) and every

x \in X.

Using

u := -1

shows that every balanced function is a symmetric function. A real-valued function

p : X \to \R

is a seminorm if and only if it is a balanced sublinear function.

References

Proofs

Sources

* * * * * * * * * * * * * * * {{TopologicalVectorSpaces Linear algebra

Definition

Balanced core

Examples

Sufficient conditions

Balanced neighborhoods

Properties

Related notions

See also

References

Sources