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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 \mathbb 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 , \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 set 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 (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 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_ or B_r for some 0 \leq r \leq \infty. Balanced set A subset S of X is called a ' or ''balanced'' if it satisfies any of the following equivalent conditions:
  1. ''Definition'': a s \in S for all s \in S and all scalars a satisfying , a, \leq 1.
  2. a S \subseteq S for all scalars a satisfying , a, \leq 1.
  3. B_ S \subseteq S (where B_ := \).
  4. S = B_ S.
  5. 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, S \cap \mathbb s is a balanced set (according to any of the previous defining conditions).
  6. 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).
  7. 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.
  8. S is a balanced subset of \operatorname S (according to any defining condition of "balanced" other than this one). * Thus S is a balanced subset of X if and only if it is balanced subset of every (equivalently, of some) vector space over the field \mathbb that contains S. So assuming that the field \mathbb is clear from context, this justifies writing "S is balanced" without mentioning any vector space.Assuming that all vector spaces containing a set S are over the same field, when describing the set as being "balanced", it is not necessary to mention a vector space containing S. That is, "S is balanced" may be written in place of "S is a balanced subset of X".
If S is a convex set then this list may be extended to include:
  1. a S \subseteq S for all scalars a satisfying , a, = 1.
If \mathbb = \R then this list may be extended to include:
  1. S is symmetric (meaning - S = S) and intersection of all balanced sets containing S.
  2. \operatorname S = \bigcup_ (a S).
  3. \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:
  1. ''Definition'': \operatorname S is the largest (with respect to \,\subseteq\,) balanced subset of S.
  2. \operatorname S is the union of all balanced subsets of S.
  3. \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 (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 p 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 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 (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 ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 for ...
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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
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:
  1. \varnothing
  2. X
  3. \
  4. \ for some real r > 0
  5. \ for some real r > 0.
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 \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 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. Nor is B is absorbing (despite the fact that \operatorname B = \R^2 is the entire vector space). 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 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 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\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 triangles 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 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 \mathbb).


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 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 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 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. Similarly for real vector spaces, if T denotes the convex hull of (0, 0) and (\pm 1, 1) (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 B := T \cup (-T) is an ( hour glass shaped) balanced subset of X := \Reals^2 whose non-empty topological interior does not contain the origin and so is not a balanced set (and although the set \ \cup \operatorname_X B 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 ...
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 \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 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, ...
p_D : X \to \R of D 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 X, thereby making \left(X, p_D\right) into a seminormed space 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 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 := u W; if in addition D does not contain any non-trivial vector subspace then 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 (, ) 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 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 and balanced. Every balanced set is star-shaped (at 0) and a symmetric set. If B is a balanced subset of X then: 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 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:
  1. p(a x) \leq p(x) whenever a is a scalar satisfying , a, \leq 1 and x \in X.
  2. p(a x) \leq p(b x) whenever a and b are scalars satisfying , a, \leq , b, and x \in X.
  3. \ 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 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

* * * * * * *


References

Proofs


Sources

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