{"history":[],"article":{"id":"89fb6383-9fd7-4c93-bebd-fead767de565","title":"Balanced set","slug":"balanced-set","content":":::summary\n:::\n\n# \nIn [linear algebra](https://bliptext.com/articles/linear-algebra) and related areas of [mathematics](https://bliptext.com/articles/mathematics) a **balanced [set](https://bliptext.com/articles/set-mathematics)**, **circled [set](https://bliptext.com/articles/set-mathematics)** or **disk** in a [vector space](https://bliptext.com/articles/vector-space) (over a [field](https://bliptext.com/articles/field-mathematics) \\mathbb{K} with an [absolute value](https://bliptext.com/articles/absolute-value-algebra) function |\\cdot |) is a [set](https://bliptext.com/articles/set-mathematics) S such that for all [scalars](https://bliptext.com/articles/scalar-mathematics) a satisfying |a| \\leq 1.\nThe **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.\nBalanced sets are ubiquitous in [functional analysis](https://bliptext.com/articles/functional-analysis) because every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin in every [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) contains a balanced [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin and every convex [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin contains a balanced convex [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin (even if the TVS is not [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space)).\nThis neighborhood can also be chosen to be an [open set](https://bliptext.com/articles/open-set) or, alternatively, a [closed set](https://bliptext.com/articles/closed-set).\n\n# Definition\nLet X be a vector space over the [field](https://bliptext.com/articles/field-mathematics) \\mathbb{K} of [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) numbers.\n**Notation**\nIf S is a set, a is a scalar, and then let and and for any let\ndenote, respectively, the open ball and the closed ball of radius r in the scalar field \\mathbb{K} centered at 0 where and\nEvery balanced subset of the field \\mathbb{K} is of the form B_{\\leq r} or B_r for some\n**Balanced set**\nA subset S of X is called a or balanced if it satisfies any of the following equivalent conditions:\n

Definition: a s \\in S for all s \\in S and all scalars a satisfying |a| \\leq 1.
for all scalars a satisfying |a| \\leq 1.
(where ).
For every s \\in S,
For every 1-dimensional vector subspace Y of S \\cap Y is a balanced set (according to any defining condition other than this one).
For every s \\in S, there exists some such that or
S is a balanced subset of (according to any defining condition of \"balanced\" other than this one).\n

\nIf S is a [convex set](https://bliptext.com/articles/convex-set) then this list may be extended to include:\n

for all scalars a satisfying |a| = 1.

\n\nIf then this list may be extended to include:\n

S is [symmetric](https://bliptext.com/articles/symmetric-set) (meaning - S = S) and

\n\n\n## Balanced hull\nThe of a subset S of X, denoted by is defined in any of the following equivalent ways:\n

Definition: is the smallest (with respect to ) balanced subset of X containing S.
is the [intersection](https://bliptext.com/articles/intersection-set-theory) of all balanced sets containing S.

\n\n## Balanced core\nThe of a subset S of X, denoted by is defined in any of the following equivalent ways:\n

Definition: is the largest (with respect to ) balanced subset of S.
is the union of all balanced subsets of S.
if 0 \\not\\in S while if 0 \\in S.

\n\n# Examples\nThe [empty set](https://bliptext.com/articles/empty-set) is a balanced set.\nAs is any vector subspace of any (real or complex) [vector space](https://bliptext.com/articles/vector-space).\nIn particular, \\{0\\} is always a balanced set.\nAny non-empty set that does not contain the origin is not balanced and furthermore, the [balanced core](https://bliptext.com/articles/) of such a set will equal the empty set.\n**Normed and topological vector spaces**\nThe open and closed [balls](https://bliptext.com/articles/open-ball) centered at the origin in a [normed vector space](https://bliptext.com/articles/normed-vector-space) are balanced sets.\nIf p is a [seminorm](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) (or [norm](https://bliptext.com/articles/norm-mathematics)) on a vector space X then for any constant c > 0, the set is balanced.\nIf is any subset and then B_1 S is a balanced set.\nIn particular, if is any balanced [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin in a [topological vector space](https://bliptext.com/articles/topological-vector-space) X then\n**Balanced sets in \\R and \\Complex**\nLet \\mathbb{K} be the field [real numbers](https://bliptext.com/articles/real-number) \\R or [complex numbers](https://bliptext.com/articles/complex-number) \\Complex, let |\\cdot| denote the [absolute value](https://bliptext.com/articles/absolute-value) on \\mathbb{K}, and let denotes the vector space over \\mathbb{K}.\nSo 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.\nThe balanced subsets of are exactly the following:\n

\\varnothing
X
\\{0\\}
for some real r > 0
for some real r > 0.

\nConsequently, both the [balanced core](https://bliptext.com/articles/) and the [balanced hull](https://bliptext.com/articles/) of every set of scalars is equal to one of the sets listed above.\nThe balanced sets are \\Complex itself, the empty set and the open and closed discs centered at zero.\nContrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at the origin will do.\nAs a result, \\Complex and \\R^2 are entirely different as far as [scalar multiplication](https://bliptext.com/articles/scalar-multiplication) is concerned.\n**Balanced sets in \\R^2**\nThroughout, let X = \\R^2 (so X is a vector space over \\R) and let B_{\\leq 1} is the closed unit ball in X centered at the origin.\nIf is non-zero, and then the set is a closed, symmetric, and balanced neighborhood of the origin in X. More generally, if C is closed subset of X such that then 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.\nLet 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).\nThen B is balanced but not convex.\nNor is B is absorbing (despite the fact that is the entire vector space).\nFor every 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 and Then the set is a balanced and absorbing set but it is not necessarily convex.\nThe [balanced hull](https://bliptext.com/articles/) of a closed set need not be closed.\nTake for instance the graph of x y = 1 in X = \\R^2.\nThe next example shows that the [balanced hull](https://bliptext.com/articles/) of a convex set may fail to be convex (however, the convex hull of a balanced set is always balanced).\nFor an example, let the convex subset be which is a horizontal closed line segment lying above the x-axis in X := \\R^2.\nThe balanced hull is a non-convex subset that is \"[hour glass](https://bliptext.com/articles/hour-glass) shaped\" and equal to the union of two closed and filled [isosceles triangles](https://bliptext.com/articles/isosceles-triangle) 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](https://bliptext.com/articles/convex-hull) of while T_2 is the [convex hull](https://bliptext.com/articles/convex-hull) of ).\n\n## Sufficient conditions\nA set T 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:\nThe [Cartesian product](https://bliptext.com/articles/cartesian-product) of a family of balanced sets is balanced in the [product space](https://bliptext.com/articles/product-space) of the corresponding vector spaces (over the same field \\mathbb{K}).\n

The balanced hull of a [compact](https://bliptext.com/articles/compact-space) (respectively, [totally bounded](https://bliptext.com/articles/totally-[bounded](https://bliptext.com/articles/bounded-set-topological-vector-space)), [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space)) set has the same property.
The convex hull of a balanced set is convex and balanced (that is, it is [absolutely convex](https://bliptext.com/articles/absolutely-convex-set)).\nHowever, the balanced hull of a convex set may fail to be convex (a counter-example is given above).
Arbitrary [unions](https://bliptext.com/articles/union-set-theory) of balanced sets are balanced, and the same is true of arbitrary [intersections](https://bliptext.com/articles/intersection-set-theory) of balanced sets.
Scalar multiples and (finite) [Minkowski sums](https://bliptext.com/articles/minkowski-sum) of balanced sets are again balanced.
Images and preimages of balanced sets under [linear maps](https://bliptext.com/articles/linear-map) are again balanced.\nExplicitly, if L : X \\to Y is a linear map and and are balanced sets, then L(B) and L^{-1}(C) are balanced sets.

\n\n## Balanced neighborhoods\nIn any [topological vector space](https://bliptext.com/articles/topological-vector-space), the closure of a balanced set is balanced.\nThe union of the origin \\{0\\} and the [topological interior](https://bliptext.com/articles/topological-interior) of a balanced set is balanced.\nTherefore, the topological interior of a balanced [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin is balanced.\nHowever, 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.\nSimilarly for real vector spaces, if T denotes the convex hull of (0, 0) and (\\pm 1, 1) (a filled [triangle](https://bliptext.com/articles/triangle) whose vertices are these three points) then is an ([hour glass](https://bliptext.com/articles/hour-glass) shaped) balanced subset of whose non-empty topological interior does not contain the origin and so is not a balanced set (and although the set formed by adding the origin is balanced, it is neither an open set nor a neighborhood of the origin).\nEvery neighborhood (respectively, convex neighborhood) of the origin in a [topological vector space](https://bliptext.com/articles/topological-vector-space) X contains a balanced (respectively, convex and balanced) open neighborhood of the origin.\nIn fact, the following construction produces such balanced sets.\nGiven the [symmetric set](https://bliptext.com/articles/symmetric-set) will be convex (respectively, closed, balanced, [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space), a neighborhood of the origin, an [absorbing subset](https://bliptext.com/articles/absorbing-set) of X) whenever this is true of W. It will be a balanced set if W is a [star shaped](https://bliptext.com/articles/star-domain) at the origin, which is true, for instance, when W is convex and contains 0.\nIn particular, if W is a convex[ neighborhood](https://bliptext.com/articles/open-neighborhood) of the origin then will be a convex[ neighborhood](https://bliptext.com/articles/open-neighborhood) of the origin and so its [topological interior](https://bliptext.com/articles/topological-interior) will be a balanced convex [ neighborhood](https://bliptext.com/articles/open-neighborhood) of the origin.\nSuppose that W is a convex and [absorbing subset](https://bliptext.com/articles/absorbing-set) of X. Then will be [convex balanced](https://bliptext.com/articles/absolutely-convex-set) [absorbing subset](https://bliptext.com/articles/absorbing-set) of X, which guarantees that the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) of D will be a [seminorm](https://bliptext.com/articles/seminorm) on X, thereby making into a [seminorm](https://bliptext.com/articles/seminorm)ed space that carries its canonical [pseduometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) topology.\nThe set of scalar multiples r D as r ranges over (or over any other set of non-zero scalars having 0 as a limit point) forms a neighborhood basis of absorbing [disks](https://bliptext.com/articles/absolutely-convex-set) at the origin for this [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) topology.\nIf X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) and if this convex absorbing subset W is also a [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of X, then the same will be true of the absorbing disk if in addition D does not contain any non-trivial vector subspace then p_D will be a [norm](https://bliptext.com/articles/norm-mathematics) and will form what is known as an auxiliary [norm](https://bliptext.com/articles/norm-mathematics)ed space.\nIf this normed space is a [Banach space](https://bliptext.com/articles/banach-space) then D is called a.\n\n# Properties\n**Properties of balanced sets**\nA balanced set is not empty if and only if it contains the origin.\nBy definition, a set is [absolutely convex](https://bliptext.com/articles/absolutely-[convex](https://bliptext.com/articles/convex-set)) if and only if it is [convex](https://bliptext.com/articles/convex-set) and balanced.\nEvery balanced set is [star-shaped](https://bliptext.com/articles/star-domain) (at 0) and a [symmetric set](https://bliptext.com/articles/symmetric-set).\nIf B is a balanced subset of X then:\n

for any scalars c and d, if then and Thus if c and d are any scalars then
B is [absorbing](https://bliptext.com/articles/absorbing-set) 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.\nIf B is not empty and if Y is a 1-dimensional vector subspace of then B \\cap Y is either \\{0\\} or else it is [absorbing](https://bliptext.com/articles/absorbing-set) in Y.
for any x \\in X, if contains more than one point then it is a convex and balanced neighborhood of 0 in the 1-dimensional vector space when this space is endowed with the [Hausdorff](https://bliptext.com/articles/hausdorff-space) [Euclidean topology](https://bliptext.com/articles/euclidean-topology); and the set B \\cap \\R x is a convex balanced subset of the real vector space \\R x that contains the origin.

\n**Properties of balanced hulls and balanced cores**\nFor any collection \\mathcal{S} of subsets of X,\nIn any topological vector space, the [balanced hull](https://bliptext.com/articles/) of any open neighborhood of the origin is again open.\nIf X is a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) and if K is a compact subset of X then the balanced hull of K is compact.\nIf a set is closed (respectively, convex, [absorbing](https://bliptext.com/articles/absorbing-set), a neighborhood of the origin) then the same is true of its balanced core.\nFor any subset and any scalar c,\nFor any scalar c \\neq 0, This equality holds for c = 0 if and only if Thus if 0 \\in S or then for every scalar c.\n\n# Related notions\nA function on a real or complex vector space is said to be a if it satisfies any of the following equivalent conditions:\n

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

\nIf p is a balanced function then for every scalar a and vector x \\in X; so in particular, for every [unit length](https://bliptext.com/articles/unit-length) scalar u (satisfying |u| = 1) and every x \\in X. Using u := -1 shows that every balanced function is a [symmetric function](https://bliptext.com/articles/symmetric-function).\nA real-valued function is a [seminorm](https://bliptext.com/articles/seminorm) if and only if it is a balanced [sublinear function](https://bliptext.com/articles/sublinear-function).\n\n## Sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:35:39.915Z","updated_at":"2025-01-25T21:35:39.915Z","search_vector":"'balanc':1 'set':2","search_content":" \n \n\n# \nIn [linear algebra](https://bliptext.com/articles/linear-algebra) and related areas of [mathematics](https://bliptext.com/articles/mathematics) a **balanced [set](https://bliptext.com/articles/set-mathematics)**, **circled [set](https://bliptext.com/articles/set-mathematics)** or **disk** in a [vector space](https://bliptext.com/articles/vector-space) (over a [field](https://bliptext.com/articles/field-mathematics) \\mathbb{K} with an [absolute value](https://bliptext.com/articles/absolute-value-algebra) function |\\cdot |) is a [set](https://bliptext.com/articles/set-mathematics) S such that for all [scalars](https://bliptext.com/articles/scalar-mathematics) a satisfying |a| \\leq 1.\nThe **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.\nBalanced sets are ubiquitous in [functional analysis](https://bliptext.com/articles/functional-analysis) because every [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin in every [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) contains a balanced [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin and every convex [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin contains a balanced convex [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin (even if the TVS is not [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space)).\nThis neighborhood can also be chosen to be an [open set](https://bliptext.com/articles/open-set) or, alternatively, a [closed set](https://bliptext.com/articles/closed-set).\n\n# Definition\nLet X be a vector space over the [field](https://bliptext.com/articles/field-mathematics) \\mathbb{K} of [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) numbers.\n**Notation**\nIf S is a set, a is a scalar, and then let and and for any let\ndenote, respectively, the open ball and the closed ball of radius r in the scalar field \\mathbb{K} centered at 0 where and\nEvery balanced subset of the field \\mathbb{K} is of the form B_{\\leq r} or B_r for some\n**Balanced set**\nA subset S of X is called a or balanced if it satisfies any of the following equivalent conditions:\n \n Definition: a s \\in S for all s \\in S and all scalars a satisfying |a| \\leq 1. \n for all scalars a satisfying |a| \\leq 1. \n (where ). \n \n For every s \\in S, For every 1-dimensional vector subspace Y of S \\cap Y is a balanced set (according to any defining condition other than this one). \n For every s \\in S, there exists some such that or \n S is a balanced subset of (according to any defining condition of \"balanced\" other than this one).\n \nIf S is a [convex set](https://bliptext.com/articles/convex-set) then this list may be extended to include:\n for all scalars a satisfying |a| = 1. \n \nIf then this list may be extended to include:\n S is [symmetric](https://bliptext.com/articles/symmetric-set) (meaning - S = S) and \n \n\n## Balanced hull\nThe of a subset S of X, denoted by is defined in any of the following equivalent ways:\n \n Definition: is the smallest (with respect to ) balanced subset of X containing S. \n is the [intersection](https://bliptext.com/articles/intersection-set-theory) of all balanced sets containing S. \n \n \n \n\n## Balanced core\nThe of a subset S of X, denoted by is defined in any of the following equivalent ways:\n \n Definition: is the largest (with respect to ) balanced subset of S. \n is the union of all balanced subsets of S. \n if 0 \\not\\in S while if 0 \\in S. \n \n\n# Examples\nThe [empty set](https://bliptext.com/articles/empty-set) is a balanced set.\nAs is any vector subspace of any (real or complex) [vector space](https://bliptext.com/articles/vector-space).\nIn particular, \\{0\\} is always a balanced set.\nAny non-empty set that does not contain the origin is not balanced and furthermore, the [balanced core](https://bliptext.com/articles/) of such a set will equal the empty set.\n**Normed and topological vector spaces**\nThe open and closed [balls](https://bliptext.com/articles/open-ball) centered at the origin in a [normed vector space](https://bliptext.com/articles/normed-vector-space) are balanced sets.\nIf p is a [semi[norm](https://bliptext.com/articles/norm-mathematics)](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) (or [norm](https://bliptext.com/articles/norm-mathematics)) on a vector space X then for any constant c > 0, the set is balanced.\nIf is any subset and then B_1 S is a balanced set.\nIn particular, if is any balanced [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin in a [topological vector space](https://bliptext.com/articles/topological-vector-space) X then\n**Balanced sets in \\R and \\Complex**\nLet \\mathbb{K} be the field [real numbers](https://bliptext.com/articles/real-number) \\R or [complex numbers](https://bliptext.com/articles/complex-number) \\Complex, let |\\cdot| denote the [absolute value](https://bliptext.com/articles/absolute-value) on \\mathbb{K}, and let denotes the vector space over \\mathbb{K}.\nSo 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.\nThe balanced subsets of are exactly the following:\n \n \\varnothing \n X \n \\{0\\} \n for some real r > 0 \n for some real r > 0. \n \nConsequently, both the [balanced core](https://bliptext.com/articles/) and the [balanced hull](https://bliptext.com/articles/) of every set of scalars is equal to one of the sets listed above.\nThe balanced sets are \\Complex itself, the empty set and the open and closed discs centered at zero.\nContrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at the origin will do.\nAs a result, \\Complex and \\R^2 are entirely different as far as [scalar multiplication](https://bliptext.com/articles/scalar-multiplication) is concerned.\n**Balanced sets in \\R^2**\nThroughout, let X = \\R^2 (so X is a vector space over \\R) and let B_{\\leq 1} is the closed unit ball in X centered at the origin.\nIf is non-zero, and then the set is a closed, symmetric, and balanced neighborhood of the origin in X. More generally, if C is closed subset of X such that then 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.\nLet 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).\nThen B is balanced but not convex.\nNor is B is absorbing (despite the fact that is the entire vector space).\nFor every 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 and Then the set is a balanced and absorbing set but it is not necessarily convex.\nThe [balanced hull](https://bliptext.com/articles/) of a closed set need not be closed.\nTake for instance the graph of x y = 1 in X = \\R^2.\nThe next example shows that the [balanced hull](https://bliptext.com/articles/) of a convex set may fail to be convex (however, the convex hull of a balanced set is always balanced).\nFor an example, let the convex subset be which is a horizontal closed line segment lying above the x-axis in X := \\R^2.\nThe balanced hull is a non-convex subset that is \"[hour glass](https://bliptext.com/articles/hour-glass) shaped\" and equal to the union of two closed and filled [isosceles triangles](https://bliptext.com/articles/isosceles-triangle) 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](https://bliptext.com/articles/convex-hull) of while T_2 is the [convex hull](https://bliptext.com/articles/convex-hull) of ).\n\n## Sufficient conditions\nA set T 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:\nThe [Cartesian product](https://bliptext.com/articles/cartesian-product) of a family of balanced sets is balanced in the [product space](https://bliptext.com/articles/product-space) of the corresponding vector spaces (over the same field \\mathbb{K}).\n \n The balanced hull of a [compact](https://bliptext.com/articles/compact-space) (respectively, [totally [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space)](https://bliptext.com/articles/totally-[bounded](https://bliptext.com/articles/bounded-set-topological-vector-space)), [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space)) set has the same property. \n The convex hull of a balanced set is convex and balanced (that is, it is [absolutely convex](https://bliptext.com/articles/absolutely-convex-set)).\nHowever, the balanced hull of a convex set may fail to be convex (a counter-example is given above). \n Arbitrary [unions](https://bliptext.com/articles/union-set-theory) of balanced sets are balanced, and the same is true of arbitrary [intersections](https://bliptext.com/articles/intersection-set-theory) of balanced sets. \n Scalar multiples and (finite) [Minkowski sums](https://bliptext.com/articles/minkowski-sum) of balanced sets are again balanced. \n Images and preimages of balanced sets under [linear maps](https://bliptext.com/articles/linear-map) are again balanced.\nExplicitly, if L : X \\to Y is a linear map and and are balanced sets, then L(B) and L^{-1}(C) are balanced sets. \n \n\n## Balanced neighborhoods\nIn any [topological vector space](https://bliptext.com/articles/topological-vector-space), the closure of a balanced set is balanced.\nThe union of the origin \\{0\\} and the [topological interior](https://bliptext.com/articles/topological-interior) of a balanced set is balanced.\nTherefore, the topological interior of a balanced [neighborhood](https://bliptext.com/articles/neighbourhood-mathematics) of the origin is balanced.\nHowever, 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.\nSimilarly for real vector spaces, if T denotes the convex hull of (0, 0) and (\\pm 1, 1) (a filled [triangle](https://bliptext.com/articles/triangle) whose vertices are these three points) then is an ([hour glass](https://bliptext.com/articles/hour-glass) shaped) balanced subset of whose non-empty topological interior does not contain the origin and so is not a balanced set (and although the set formed by adding the origin is balanced, it is neither an open set nor a neighborhood of the origin).\nEvery neighborhood (respectively, convex neighborhood) of the origin in a [topological vector space](https://bliptext.com/articles/topological-vector-space) X contains a balanced (respectively, convex and balanced) open neighborhood of the origin.\nIn fact, the following construction produces such balanced sets.\nGiven the [symmetric set](https://bliptext.com/articles/symmetric-set) will be convex (respectively, closed, balanced, [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space), a neighborhood of the origin, an [absorbing subset](https://bliptext.com/articles/absorbing-set) of X) whenever this is true of W. It will be a balanced set if W is a [star shaped](https://bliptext.com/articles/star-domain) at the origin, which is true, for instance, when W is convex and contains 0.\nIn particular, if W is a convex[ neighborhood](https://bliptext.com/articles/open-neighborhood) of the origin then will be a convex[ neighborhood](https://bliptext.com/articles/open-neighborhood) of the origin and so its [topological interior](https://bliptext.com/articles/topological-interior) will be a balanced convex [ neighborhood](https://bliptext.com/articles/open-neighborhood) of the origin.\nSuppose that W is a convex and [absorbing subset](https://bliptext.com/articles/absorbing-set) of X. Then will be [convex balanced](https://bliptext.com/articles/absolutely-convex-set) [absorbing subset](https://bliptext.com/articles/absorbing-set) of X, which guarantees that the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) of D will be a [seminorm](https://bliptext.com/articles/seminorm) on X, thereby making into a [seminorm](https://bliptext.com/articles/seminorm)ed space that carries its canonical [pseduometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) topology.\nThe set of scalar multiples r D as r ranges over (or over any other set of non-zero scalars having 0 as a limit point) forms a neighborhood basis of absorbing [disks](https://bliptext.com/articles/absolutely-convex-set) at the origin for this [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) topology.\nIf X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) and if this convex absorbing subset W is also a [bounded subset](https://bliptext.com/articles/bounded-set-topological-vector-space) of X, then the same will be true of the absorbing disk if in addition D does not contain any non-trivial vector subspace then p_D will be a [norm](https://bliptext.com/articles/norm-mathematics) and will form what is known as an auxiliary [norm](https://bliptext.com/articles/norm-mathematics)ed space.\nIf this normed space is a [Banach space](https://bliptext.com/articles/banach-space) then D is called a.\n\n# Properties\n**Properties of balanced sets**\nA balanced set is not empty if and only if it contains the origin.\nBy definition, a set is [absolutely [convex](https://bliptext.com/articles/convex-set)](https://bliptext.com/articles/absolutely-[convex](https://bliptext.com/articles/convex-set)) if and only if it is [convex](https://bliptext.com/articles/convex-set) and balanced.\nEvery balanced set is [star-shaped](https://bliptext.com/articles/star-domain) (at 0) and a [symmetric set](https://bliptext.com/articles/symmetric-set).\nIf B is a balanced subset of X then:\n \n for any scalars c and d, if then and Thus if c and d are any scalars then \n B is [absorbing](https://bliptext.com/articles/absorbing-set) in X if and only if for all x \\in X, there exists r > 0 such that x \\in r B. \n for any 1-dimensional vector subspace Y of X, the set B \\cap Y is convex and balanced.\nIf B is not empty and if Y is a 1-dimensional vector subspace of then B \\cap Y is either \\{0\\} or else it is [absorbing](https://bliptext.com/articles/absorbing-set) in Y. \n for any x \\in X, if contains more than one point then it is a convex and balanced neighborhood of 0 in the 1-dimensional vector space when this space is endowed with the [Hausdorff](https://bliptext.com/articles/hausdorff-space) [Euclidean topology](https://bliptext.com/articles/euclidean-topology); and the set B \\cap \\R x is a convex balanced subset of the real vector space \\R x that contains the origin. \n \n**Properties of balanced hulls and balanced cores**\nFor any collection \\mathcal{S} of subsets of X,\nIn any topological vector space, the [balanced hull](https://bliptext.com/articles/) of any open neighborhood of the origin is again open.\nIf X is a [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) and if K is a compact subset of X then the balanced hull of K is compact.\nIf a set is closed (respectively, convex, [absorbing](https://bliptext.com/articles/absorbing-set), a neighborhood of the origin) then the same is true of its balanced core.\nFor any subset and any scalar c,\nFor any scalar c \\neq 0, This equality holds for c = 0 if and only if Thus if 0 \\in S or then for every scalar c.\n\n# Related notions\nA function on a real or complex vector space is said to be a if it satisfies any of the following equivalent conditions:\n \n whenever a is a scalar satisfying |a| \\leq 1 and x \\in X. \n whenever a and b are scalars satisfying and x \\in X. \n is a balanced set for every non-negative real t \\geq 0. \n \nIf p is a balanced function then for every scalar a and vector x \\in X; so in particular, for every [unit length](https://bliptext.com/articles/unit-length) scalar u (satisfying |u| = 1) and every x \\in X. Using u := -1 shows that every balanced function is a [symmetric function](https://bliptext.com/articles/symmetric-function).\nA real-valued function is a [seminorm](https://bliptext.com/articles/seminorm) if and only if it is a balanced [sublinear function](https://bliptext.com/articles/sublinear-function).\n\n## Sources","revisionCount":0},"hasMore":false,"nextCursor":null}

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