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 matrices.
...
and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 can ...
(over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
function
) is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
such that
for all
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
s
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 space#Definition, inner product, Norm (mathematics)#Defini ...
because every
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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
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 ve ...
). 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 are suf ...
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 space, a cl ...
.
Definition
Let
be a vector space over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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 r ...
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 definiti ...
(meaning ) and
- \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 of
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)
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 can ...
. 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 "length" i ...
are balanced sets. If
p is a
seminorm 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 ...
(or
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
) 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
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 ...
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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s
\R 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 form ...
s
\Complex, let
, \cdot, 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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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 by ...
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
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
x y = 1 in
X = \R^2.
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
S := 1, 1
Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...
\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
An hourglass (or sandglass, sand timer, sand clock or egg timer) is a device used to measure the passage of time. It comprises two glass bulbs connected vertically by a narrow neck that allows a regulated flow of a substance (historically sand) ...
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
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
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
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\ti ...
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-seemin ...
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 “size ...
, bounded) set has the same property.
- The convex hull of a balanced set is convex and balanced (that is, it is
absolutely convex In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
The disked hull ...
). 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 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 Minkowski ...
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 Map (mathematics), mapping V \to W between two vect ...
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
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
\ and the
topological interior of a balanced set is balanced. Therefore, the topological interior of a balanced
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 isomorphi ...
, 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
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 ...
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 to ...
\bigcap_ u W \subseteq W will be convex (respectively, closed, balanced,
bounded, 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 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 p_D : X \to \R of
D will be a
seminorm 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 ...
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 and ...
that carries its canonical
pseduometrizable topology. The set of scalar multiples
r D as
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
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 ve ...
topology. If
X 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
W is also a
bounded subset of
X, then the same will be true of the absorbing disk
D := \bigcap_ u W, in which case
p_D will be a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
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 vector ...
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 In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
The disked hull ...
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 polytope ...
and balanced.
Every balanced set is
star-shaped (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 to ...
.
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 distance, Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative f ...
; 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
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
X 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
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 or
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.
If
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 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 ...
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 seminorm. ...
.
See also
*
*
*
*
*
*
*
References
Proofs
Sources
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
{{TopologicalVectorSpaces
Linear algebra