• Images
• Videos
• Documents
• Books
• Articles

Definition

Let X {\displaystyle X} be a vector space over the field K {\displaystyle \mathbb {K} } of real or complex numbers.

Notation

If S {\displaystyle S} is a set, a {\displaystyle a} is a scalar, and B ⊆ K {\displaystyle B\subseteq \mathbb {K} } then let a S = { a s : s ∈ S } {\displaystyle aS=\{as:s\in S\}} and B S = { b s : b ∈ B , s ∈ S } {\displaystyle BS=\{bs:b\in B,s\in S\}} and for any 0 ≤ r ≤ ∞ , {\displaystyle 0\leq r\leq \infty ,} let B r = { a ∈ K : | a | < r }  and  B ≤ r = { a ∈ K : | a | ≤ r } . {\displaystyle B_{r}=\{a\in \mathbb {K} :|a|<r\}\qquad {\text{ and }}\qquad B_{\leq r}=\{a\in \mathbb {K} :|a|\leq r\}.} denote, respectively, the open ball and the closed ball of radius r {\displaystyle r} in the scalar field K {\displaystyle \mathbb {K} } centered at 0 {\displaystyle 0} where B 0 = ∅ , B ≤ 0 = { 0 } , {\displaystyle B_{0}=\varnothing ,B_{\leq 0}=\{0\},} and B ∞ = B ≤ ∞ = K . {\displaystyle B_{\infty }=B_{\leq \infty }=\mathbb {K} .} Every balanced subset of the field K {\displaystyle \mathbb {K} } is of the form B ≤ r {\displaystyle B_{\leq r}} or B r {\displaystyle B_{r}} for some 0 ≤ r ≤ ∞ . {\displaystyle 0\leq r\leq \infty .}

Balanced set

A subset S {\displaystyle S} of X {\displaystyle X} is called a balanced set or balanced if it satisfies any of the following equivalent conditions:

1. Definition: a s ∈ S {\displaystyle as\in S} for all s ∈ S {\displaystyle s\in S} and all scalars a {\displaystyle a} satisfying | a | ≤ 1. {\displaystyle |a|\leq 1.}
2. a S ⊆ S {\displaystyle aS\subseteq S} for all scalars a {\displaystyle a} satisfying | a | ≤ 1. {\displaystyle |a|\leq 1.}
3. B ≤ 1 S ⊆ S {\displaystyle B_{\leq 1}S\subseteq S} (where B ≤ 1 := { a ∈ K : | a | ≤ 1 } {\displaystyle B_{\leq 1}:=\{a\in \mathbb {K} :|a|\leq 1\}} ).
4. S = B ≤ 1 S . {\displaystyle S=B_{\leq 1}S.} ¹
5. For every s ∈ S , {\displaystyle s\in S,} S ∩ K s = B ≤ 1 ( S ∩ K s ) . {\displaystyle S\cap \mathbb {K} s=B_{\leq 1}(S\cap \mathbb {K} s).}
   • K s = span ⁡ { s } {\displaystyle \mathbb {K} s=\operatorname {span} \{s\}} is a 0 {\displaystyle 0} (if s = 0 {\displaystyle s=0} ) or 1 {\displaystyle 1} (if s ≠ 0 {\displaystyle s\neq 0} ) dimensional vector subspace of X . {\displaystyle X.}
   • If R := S ∩ K s {\displaystyle R:=S\cap \mathbb {K} s} then the above equality becomes R = B ≤ 1 R , {\displaystyle R=B_{\leq 1}R,} which is exactly the previous condition for a set to be balanced. Thus, S {\displaystyle S} is balanced if and only if for every s ∈ S , {\displaystyle s\in S,} S ∩ K s {\displaystyle S\cap \mathbb {K} s} is a balanced set (according to any of the previous defining conditions).
6. For every 1-dimensional vector subspace Y {\displaystyle Y} of span ⁡ S , {\displaystyle \operatorname {span} S,} S ∩ Y {\displaystyle S\cap Y} is a balanced set (according to any defining condition other than this one).
7. For every s ∈ S , {\displaystyle s\in S,} there exists some 0 ≤ r ≤ ∞ {\displaystyle 0\leq r\leq \infty } such that S ∩ K s = B r s {\displaystyle S\cap \mathbb {K} s=B_{r}s} or S ∩ K s = B ≤ r s . {\displaystyle S\cap \mathbb {K} s=B_{\leq r}s.}
8. S {\displaystyle S} is a balanced subset of span ⁡ S {\displaystyle \operatorname {span} S} (according to any defining condition of "balanced" other than this one).
   • Thus S {\displaystyle S} is a balanced subset of X {\displaystyle X} if and only if it is balanced subset of every (equivalently, of some) vector space over the field K {\displaystyle \mathbb {K} } that contains S . {\displaystyle S.} So assuming that the field K {\displaystyle \mathbb {K} } is clear from context, this justifies writing " S {\displaystyle S} is balanced" without mentioning any vector space.²

If S {\displaystyle S} is a convex set then this list may be extended to include:

1. a S ⊆ S {\displaystyle aS\subseteq S} for all scalars a {\displaystyle a} satisfying | a | = 1. {\displaystyle |a|=1.} ³

If K = R {\displaystyle \mathbb {K} =\mathbb {R} } then this list may be extended to include:

1. S {\displaystyle S} is symmetric (meaning − S = S {\displaystyle -S=S} ) and [ 0 , 1 ) S ⊆ S . {\displaystyle [0,1)S\subseteq S.}

Balanced hull

bal ⁡ S   =   ⋃ | a | ≤ 1 a S = B ≤ 1 S {\displaystyle \operatorname {bal} S~=~\bigcup _{|a|\leq 1}aS=B_{\leq 1}S}

The balanced hull of a subset S {\displaystyle S} of X , {\displaystyle X,} denoted by bal ⁡ S , {\displaystyle \operatorname {bal} S,} is defined in any of the following equivalent ways:

1. Definition: bal ⁡ S {\displaystyle \operatorname {bal} S} is the smallest (with respect to ⊆ {\displaystyle \,\subseteq \,} ) balanced subset of X {\displaystyle X} containing S . {\displaystyle S.}
2. bal ⁡ S {\displaystyle \operatorname {bal} S} is the intersection of all balanced sets containing S . {\displaystyle S.}
3. bal ⁡ S = ⋃ | a | ≤ 1 ( a S ) . {\displaystyle \operatorname {bal} S=\bigcup _{|a|\leq 1}(aS).}
4. bal ⁡ S = B ≤ 1 S . {\displaystyle \operatorname {bal} S=B_{\leq 1}S.} ⁴

Balanced core

balcore ⁡ S   =   { ⋂ | a | ≥ 1 a S  if  0 ∈ S ∅  if  0 ∉ S {\displaystyle \operatorname {balcore} S~=~{\begin{cases}\displaystyle \bigcap _{|a|\geq 1}aS&{\text{ if }}0\in S\\\varnothing &{\text{ if }}0\not \in S\\\end{cases}}}

The balanced core of a subset S {\displaystyle S} of X , {\displaystyle X,} denoted by balcore ⁡ S , {\displaystyle \operatorname {balcore} S,} is defined in any of the following equivalent ways:

1. Definition: balcore ⁡ S {\displaystyle \operatorname {balcore} S} is the largest (with respect to ⊆ {\displaystyle \,\subseteq \,} ) balanced subset of S . {\displaystyle S.}
2. balcore ⁡ S {\displaystyle \operatorname {balcore} S} is the union of all balanced subsets of S . {\displaystyle S.}
3. balcore ⁡ S = ∅ {\displaystyle \operatorname {balcore} S=\varnothing } if 0 ∉ S {\displaystyle 0\not \in S} while balcore ⁡ S = ⋂ | a | ≥ 1 ( a S ) {\displaystyle \operatorname {balcore} S=\bigcap _{|a|\geq 1}(aS)} if 0 ∈ S . {\displaystyle 0\in S.}

Examples

The empty set is a balanced set. As is any vector subspace of any (real or complex) vector space. In particular, { 0 } {\displaystyle \{0\}} 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 {\displaystyle p} is a seminorm (or norm) on a vector space X {\displaystyle X} then for any constant c > 0 , {\displaystyle c>0,} the set { x ∈ X : p ( x ) ≤ c } {\displaystyle \{x\in X:p(x)\leq c\}} is balanced.

If S ⊆ X {\displaystyle S\subseteq X} is any subset and B 1 := { a ∈ K : | a | < 1 } {\displaystyle B_{1}:=\{a\in \mathbb {K} :|a|<1\}} then B 1 S {\displaystyle B_{1}S} is a balanced set. In particular, if U ⊆ X {\displaystyle U\subseteq X} is any balanced neighborhood of the origin in a topological vector space X {\displaystyle X} then Int X ⁡ U   ⊆   B 1 U   =   ⋃ 0 < | a | < 1 a U   ⊆   U . {\displaystyle \operatorname {Int} _{X}U~\subseteq ~B_{1}U~=~\bigcup _{0<|a|<1}aU~\subseteq ~U.}

Balanced sets in R {\displaystyle \mathbb {R} } and C {\displaystyle \mathbb {C} }

Let K {\displaystyle \mathbb {K} } be the field real numbers R {\displaystyle \mathbb {R} } or complex numbers C , {\displaystyle \mathbb {C} ,} let | ⋅ | {\displaystyle |\cdot |} denote the absolute value on K , {\displaystyle \mathbb {K} ,} and let X := K {\displaystyle X:=\mathbb {K} } denotes the vector space over K . {\displaystyle \mathbb {K} .} So for example, if K := C {\displaystyle \mathbb {K} :=\mathbb {C} } is the field of complex numbers then X = K = C {\displaystyle X=\mathbb {K} =\mathbb {C} } is a 1-dimensional complex vector space whereas if K := R {\displaystyle \mathbb {K} :=\mathbb {R} } then X = K = R {\displaystyle X=\mathbb {K} =\mathbb {R} } is a 1-dimensional real vector space.

The balanced subsets of X = K {\displaystyle X=\mathbb {K} } are exactly the following:⁵

1. ∅ {\displaystyle \varnothing }
2. X {\displaystyle X}
3. { 0 } {\displaystyle \{0\}}
4. { x ∈ X : | x | < r } {\displaystyle \{x\in X:|x|<r\}} for some real r > 0 {\displaystyle r>0}
5. { x ∈ X : | x | ≤ r } {\displaystyle \{x\in X:|x|\leq r\}} for some real r > 0. {\displaystyle 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 C {\displaystyle \mathbb {C} } 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, C {\displaystyle \mathbb {C} } and R 2 {\displaystyle \mathbb {R} ^{2}} are entirely different as far as scalar multiplication is concerned.

Balanced sets in R 2 {\displaystyle \mathbb {R} ^{2}}

Throughout, let X = R 2 {\displaystyle X=\mathbb {R} ^{2}} (so X {\displaystyle X} is a vector space over R {\displaystyle \mathbb {R} } ) and let B ≤ 1 {\displaystyle B_{\leq 1}} is the closed unit ball in X {\displaystyle X} centered at the origin.

If x 0 ∈ X = R 2 {\displaystyle x_{0}\in X=\mathbb {R} ^{2}} is non-zero, and L := R x 0 , {\displaystyle L:=\mathbb {R} x_{0},} then the set R := B ≤ 1 ∪ L {\displaystyle R:=B_{\leq 1}\cup L} is a closed, symmetric, and balanced neighborhood of the origin in X . {\displaystyle X.} More generally, if C {\displaystyle C} is any closed subset of X {\displaystyle X} such that ( 0 , 1 ) C ⊆ C , {\displaystyle (0,1)C\subseteq C,} then S := B ≤ 1 ∪ C ∪ ( − C ) {\displaystyle S:=B_{\leq 1}\cup C\cup (-C)} is a closed, symmetric, and balanced neighborhood of the origin in X . {\displaystyle X.} This example can be generalized to R n {\displaystyle \mathbb {R} ^{n}} for any integer n ≥ 1. {\displaystyle n\geq 1.}

Let B ⊆ R 2 {\displaystyle B\subseteq \mathbb {R} ^{2}} be the union of the line segment between the points ( − 1 , 0 ) {\displaystyle (-1,0)} and ( 1 , 0 ) {\displaystyle (1,0)} and the line segment between ( 0 , − 1 ) {\displaystyle (0,-1)} and ( 0 , 1 ) . {\displaystyle (0,1).} Then B {\displaystyle B} is balanced but not convex. Nor is B {\displaystyle B} is absorbing (despite the fact that span ⁡ B = R 2 {\displaystyle \operatorname {span} B=\mathbb {R} ^{2}} is the entire vector space).

For every 0 ≤ t ≤ π , {\displaystyle 0\leq t\leq \pi ,} let r t {\displaystyle r_{t}} be any positive real number and let B t {\displaystyle B^{t}} be the (open or closed) line segment in X := R 2 {\displaystyle X:=\mathbb {R} ^{2}} between the points ( cos ⁡ t , sin ⁡ t ) {\displaystyle (\cos t,\sin t)} and − ( cos ⁡ t , sin ⁡ t ) . {\displaystyle -(\cos t,\sin t).} Then the set B = ⋃ 0 ≤ t < π r t B t {\displaystyle B=\bigcup _{0\leq t<\pi }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 {\displaystyle xy=1} in X = R 2 . {\displaystyle X=\mathbb {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 ] × { 1 } , {\displaystyle S:=[-1,1]\times \{1\},} which is a horizontal closed line segment lying above the x − {\displaystyle x-} axis in X := R 2 . {\displaystyle X:=\mathbb {R} ^{2}.} The balanced hull bal ⁡ S {\displaystyle \operatorname {bal} 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 {\displaystyle T_{1}} and T 2 , {\displaystyle T_{2},} where T 2 = − T 1 {\displaystyle T_{2}=-T_{1}} and T 1 {\displaystyle T_{1}} is the filled triangle whose vertices are the origin together with the endpoints of S {\displaystyle S} (said differently, T 1 {\displaystyle T_{1}} is the convex hull of S ∪ { ( 0 , 0 ) } {\displaystyle S\cup \{(0,0)\}} while T 2 {\displaystyle T_{2}} is the convex hull of ( − S ) ∪ { ( 0 , 0 ) } {\displaystyle (-S)\cup \{(0,0)\}} ).

Sufficient conditions

A set T {\displaystyle T} is balanced if and only if it is equal to its balanced hull bal ⁡ T {\displaystyle \operatorname {bal} T} or to its balanced core balcore ⁡ T , {\displaystyle \operatorname {balcore} T,} in which case all three of these sets are equal: T = bal ⁡ T = balcore ⁡ T . {\displaystyle T=\operatorname {bal} T=\operatorname {balcore} T.}

The Cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field K {\displaystyle \mathbb {K} } ).

• The balanced hull of a compact (respectively, totally bounded, bounded) set has the same property.⁶
• 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 of balanced sets.
• Scalar multiples and (finite) Minkowski sums of balanced sets are again balanced.
• Images and preimages of balanced sets under linear maps are again balanced. Explicitly, if L : X → Y {\displaystyle L:X\to Y} is a linear map and B ⊆ X {\displaystyle B\subseteq X} and C ⊆ Y {\displaystyle C\subseteq Y} are balanced sets, then L ( B ) {\displaystyle L(B)} and L − 1 ( C ) {\displaystyle L^{-1}(C)} are balanced sets.

Balanced neighborhoods

In any topological vector space, the closure of a balanced set is balanced.⁷ The union of the origin { 0 } {\displaystyle \{0\}} and the topological interior of a balanced set is balanced. Therefore, the topological interior of a balanced neighborhood of the origin is balanced.⁸⁹ However, { ( z , w ) ∈ C 2 : | z | ≤ | w | } {\displaystyle \left\{(z,w)\in \mathbb {C} ^{2}:|z|\leq |w|\right\}} is a balanced subset of X = C 2 {\displaystyle X=\mathbb {C} ^{2}} that contains the origin ( 0 , 0 ) ∈ X {\displaystyle (0,0)\in X} but whose (nonempty) topological interior does not contain the origin and is therefore not a balanced set.¹⁰ Similarly for real vector spaces, if T {\displaystyle T} denotes the convex hull of ( 0 , 0 ) {\displaystyle (0,0)} and ( ± 1 , 1 ) {\displaystyle (\pm 1,1)} (a filled triangle whose vertices are these three points) then B := T ∪ ( − T ) {\displaystyle B:=T\cup (-T)} is an (hour glass shaped) balanced subset of X := R 2 {\displaystyle X:=\mathbb {R} ^{2}} whose non-empty topological interior does not contain the origin and so is not a balanced set (and although the set { ( 0 , 0 ) } ∪ Int X ⁡ B {\displaystyle \{(0,0)\}\cup \operatorname {Int} _{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 X {\displaystyle X} contains a balanced (respectively, convex and balanced) open neighborhood of the origin. In fact, the following construction produces such balanced sets. Given W ⊆ X , {\displaystyle W\subseteq X,} the symmetric set ⋂ | u | = 1 u W ⊆ W {\displaystyle \bigcap _{|u|=1}uW\subseteq W} will be convex (respectively, closed, balanced, bounded, a neighborhood of the origin, an absorbing subset of X {\displaystyle X} ) whenever this is true of W . {\displaystyle W.} It will be a balanced set if W {\displaystyle W} is a star shaped at the origin,¹¹ which is true, for instance, when W {\displaystyle W} is convex and contains 0. {\displaystyle 0.} In particular, if W {\displaystyle W} is a convex neighborhood of the origin then ⋂ | u | = 1 u W {\displaystyle \bigcap _{|u|=1}uW} will be a balanced convex neighborhood of the origin and so its topological interior will be a balanced convex open neighborhood of the origin.¹²

Let 0 ∈ W ⊆ X {\displaystyle 0\in W\subseteq X} and define A = ⋂ | u | = 1 u W {\displaystyle A=\bigcap _{|u|=1}uW} (where u {\displaystyle u} denotes elements of the field K {\displaystyle \mathbb {K} } of scalars). Taking u := 1 {\displaystyle u:=1} shows that A ⊆ W . {\displaystyle A\subseteq W.} If W {\displaystyle W} is convex then so is A {\displaystyle A} (since an intersection of convex sets is convex) and thus so is A {\displaystyle A} 's interior. If | s | = 1 {\displaystyle |s|=1} then s A = ⋂ | u | = 1 s u W ⊆ ⋂ | u | = 1 u W = A {\displaystyle sA=\bigcap _{|u|=1}suW\subseteq \bigcap _{|u|=1}uW=A} and thus s A = A . {\displaystyle sA=A.} If W {\displaystyle W} is star shaped at the origin¹³ then so is every u W {\displaystyle uW} (for | u | = 1 {\displaystyle |u|=1} ), which implies that for any 0 ≤ r ≤ 1 , {\displaystyle 0\leq r\leq 1,} r A = ⋂ | u | = 1 r u W ⊆ ⋂ | u | = 1 u W = A {\displaystyle rA=\bigcap _{|u|=1}ruW\subseteq \bigcap _{|u|=1}uW=A} thus proving that A {\displaystyle A} is balanced. If W {\displaystyle W} is convex and contains the origin then it is star shaped at the origin and so A {\displaystyle A} will be balanced.

Now suppose W {\displaystyle W} is a neighborhood of the origin in X . {\displaystyle X.} Since scalar multiplication M : K × X → X {\displaystyle M:\mathbb {K} \times X\to X} (defined by M ( a , x ) = a x {\displaystyle M(a,x)=ax} ) is continuous at the origin ( 0 , 0 ) ∈ K × X {\displaystyle (0,0)\in \mathbb {K} \times X} and M ( 0 , 0 ) = 0 ∈ W , {\displaystyle M(0,0)=0\in W,} there exists some basic open neighborhood B r × V {\displaystyle B_{r}\times V} (where r > 0 {\displaystyle r>0} and B r := { c ∈ K : | c | < r } {\displaystyle B_{r}:=\{c\in \mathbb {K} :|c|<r\}} ) of the origin in the product topology on K × X {\displaystyle \mathbb {K} \times X} such that M ( B r × V ) ⊆ W ; {\displaystyle M\left(B_{r}\times V\right)\subseteq W;} the set M ( B r × V ) = B r V {\displaystyle M\left(B_{r}\times V\right)=B_{r}V} is balanced and it is also open because it may be written as B r V = ⋃ | a | < r a V = ⋃ 0 < | a | < r a V  (since  0 ⋅ V = { 0 } ⊆ a V  ) {\displaystyle B_{r}V=\bigcup _{|a|<r}aV=\bigcup _{0<|a|<r}aV\qquad {\text{ (since }}0\cdot V=\{0\}\subseteq aV{\text{ )}}} where a V {\displaystyle aV} is an open neighborhood of the origin whenever a ≠ 0. {\displaystyle a\neq 0.} Finally, A = ⋂ | u | = 1 u W ⊇ ⋂ | u | = 1 u B r V = ⋂ | u | = 1 B r V = B r V {\displaystyle A=\bigcap _{|u|=1}uW\supseteq \bigcap _{|u|=1}uB_{r}V=\bigcap _{|u|=1}B_{r}V=B_{r}V} shows that A {\displaystyle A} is also a neighborhood of the origin. If A {\displaystyle A} is balanced then because its interior Int X ⁡ A {\displaystyle \operatorname {Int} _{X}A} contains the origin, Int X ⁡ A {\displaystyle \operatorname {Int} _{X}A} will also be balanced. If W {\displaystyle W} is convex then A {\displaystyle A} is convex and balanced and thus the same is true of Int X ⁡ A . {\displaystyle \operatorname {Int} _{X}A.} ◼ {\displaystyle \blacksquare }

Suppose that W {\displaystyle W} is a convex and absorbing subset of X . {\displaystyle X.} Then D := ⋂ | u | = 1 u W {\displaystyle D:=\bigcap _{|u|=1}uW} will be convex balanced absorbing subset of X , {\displaystyle X,} which guarantees that the Minkowski functional p D : X → R {\displaystyle p_{D}:X\to \mathbb {R} } of D {\displaystyle D} will be a seminorm on X , {\displaystyle X,} thereby making ( X , p D ) {\displaystyle \left(X,p_{D}\right)} into a seminormed space that carries its canonical pseduometrizable topology. The set of scalar multiples r D {\displaystyle rD} as r {\displaystyle r} ranges over { 1 2 , 1 3 , 1 4 , … } {\displaystyle \left\{{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\ldots \right\}} (or over any other set of non-zero scalars having 0 {\displaystyle 0} as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If X {\displaystyle X} is a topological vector space and if this convex absorbing subset W {\displaystyle W} is also a bounded subset of X , {\displaystyle X,} then the same will be true of the absorbing disk D := ⋂ | u | = 1 u W ; {\displaystyle D:={\textstyle \bigcap \limits _{|u|=1}}uW;} if in addition D {\displaystyle D} does not contain any non-trivial vector subspace then p D {\displaystyle p_{D}} will be a norm and ( X , p D ) {\displaystyle \left(X,p_{D}\right)} will form what is known as an auxiliary normed space.¹⁴ If this normed space is a Banach space then D {\displaystyle D} is called a Banach disk.

Properties

See also: Topological vector space § 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 {\displaystyle B} is a balanced subset of X {\displaystyle X} then:

• for any scalars c {\displaystyle c} and d , {\displaystyle d,} if | c | ≤ | d | {\displaystyle |c|\leq |d|} then c B ⊆ d B {\displaystyle cB\subseteq dB} and c B = | c | B . {\displaystyle cB=|c|B.} Thus if c {\displaystyle c} and d {\displaystyle d} are any scalars then ( c B ) ∩ ( d B ) = min { | c | , | d | } B . {\displaystyle (cB)\cap (dB)=\min _{}\{|c|,|d|\}B.}
• B {\displaystyle B} is absorbing in X {\displaystyle X} if and only if for all x ∈ X , {\displaystyle x\in X,} there exists r > 0 {\displaystyle r>0} such that x ∈ r B . {\displaystyle x\in rB.} ¹⁵
• for any 1-dimensional vector subspace Y {\displaystyle Y} of X , {\displaystyle X,} the set B ∩ Y {\displaystyle B\cap Y} is convex and balanced. If B {\displaystyle B} is not empty and if Y {\displaystyle Y} is a 1-dimensional vector subspace of span ⁡ B {\displaystyle \operatorname {span} B} then B ∩ Y {\displaystyle B\cap Y} is either { 0 } {\displaystyle \{0\}} or else it is absorbing in Y . {\displaystyle Y.}
• for any x ∈ X , {\displaystyle x\in X,} if B ∩ span ⁡ x {\displaystyle B\cap \operatorname {span} x} contains more than one point then it is a convex and balanced neighborhood of 0 {\displaystyle 0} in the 1-dimensional vector space span ⁡ x {\displaystyle \operatorname {span} x} when this space is endowed with the Hausdorff Euclidean topology; and the set B ∩ R x {\displaystyle B\cap \mathbb {R} x} is a convex balanced subset of the real vector space R x {\displaystyle \mathbb {R} x} that contains the origin.

Properties of balanced hulls and balanced cores

For any collection S {\displaystyle {\mathcal {S}}} of subsets of X , {\displaystyle X,} bal ⁡ ( ⋃ S ∈ S S ) = ⋃ S ∈ S bal ⁡ S  and  balcore ⁡ ( ⋂ S ∈ S S ) = ⋂ S ∈ S balcore ⁡ S . {\displaystyle \operatorname {bal} \left(\bigcup _{S\in {\mathcal {S}}}S\right)=\bigcup _{S\in {\mathcal {S}}}\operatorname {bal} S\quad {\text{ and }}\quad \operatorname {balcore} \left(\bigcap _{S\in {\mathcal {S}}}S\right)=\bigcap _{S\in {\mathcal {S}}}\operatorname {balcore} S.}

In any topological vector space, the balanced hull of any open neighborhood of the origin is again open. If X {\displaystyle X} is a Hausdorff topological vector space and if K {\displaystyle K} is a compact subset of X {\displaystyle X} then the balanced hull of K {\displaystyle K} is compact.¹⁶

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 ⊆ X {\displaystyle S\subseteq X} and any scalar c , {\displaystyle c,} bal ⁡ ( c S ) = c bal ⁡ S = | c | bal ⁡ S . {\displaystyle \operatorname {bal} (c\,S)=c\operatorname {bal} S=|c|\operatorname {bal} S.}

For any scalar c ≠ 0 , {\displaystyle c\neq 0,} balcore ⁡ ( c S ) = c balcore ⁡ S = | c | balcore ⁡ S . {\displaystyle \operatorname {balcore} (c\,S)=c\operatorname {balcore} S=|c|\operatorname {balcore} S.} This equality holds for c = 0 {\displaystyle c=0} if and only if S ⊆ { 0 } . {\displaystyle S\subseteq \{0\}.} Thus if 0 ∈ S {\displaystyle 0\in S} or S = ∅ {\displaystyle S=\varnothing } then balcore ⁡ ( c S ) = c balcore ⁡ S = | c | balcore ⁡ S {\displaystyle \operatorname {balcore} (c\,S)=c\operatorname {balcore} S=|c|\operatorname {balcore} S} for every scalar c . {\displaystyle c.}

Related notions

A function p : X → [ 0 , ∞ ) {\displaystyle p:X\to [0,\infty )} on a real or complex vector space is said to be a balanced function if it satisfies any of the following equivalent conditions:¹⁷

1. p ( a x ) ≤ p ( x ) {\displaystyle p(ax)\leq p(x)} whenever a {\displaystyle a} is a scalar satisfying | a | ≤ 1 {\displaystyle |a|\leq 1} and x ∈ X . {\displaystyle x\in X.}
2. p ( a x ) ≤ p ( b x ) {\displaystyle p(ax)\leq p(bx)} whenever a {\displaystyle a} and b {\displaystyle b} are scalars satisfying | a | ≤ | b | {\displaystyle |a|\leq |b|} and x ∈ X . {\displaystyle x\in X.}
3. { x ∈ X : p ( x ) ≤ t } {\displaystyle \{x\in X:p(x)\leq t\}} is a balanced set for every non-negative real t ≥ 0. {\displaystyle t\geq 0.}

If p {\displaystyle p} is a balanced function then p ( a x ) = p ( | a | x ) {\displaystyle p(ax)=p(|a|x)} for every scalar a {\displaystyle a} and vector x ∈ X ; {\displaystyle x\in X;} so in particular, p ( u x ) = p ( x ) {\displaystyle p(ux)=p(x)} for every unit length scalar u {\displaystyle u} (satisfying | u | = 1 {\displaystyle |u|=1} ) and every x ∈ X . {\displaystyle x\in X.} ¹⁸ Using u := − 1 {\displaystyle u:=-1} shows that every balanced function is a symmetric function.

A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } is a seminorm if and only if it is a balanced sublinear function.

See also

• Absolutely convex set – Convex and balanced set
• Absorbing set – Set that can be "inflated" to reach any point
• Bounded set (topological vector space) – Generalization of boundedness
• Convex set – In geometry, set whose intersection with every line is a single line segment
• Star domain – Property of point sets in Euclidean spaces
• Symmetric set – Property of group subsets (mathematics)
• Topological vector space – Vector space with a notion of nearness

Proofs

Sources

• Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190.
• Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908.
• Dunford, Nelson; Schwartz, Jacob T. (1988). Linear Operators. Pure and applied mathematics. Vol. 1. New York: Wiley-Interscience. ISBN 978-0-471-60848-6. OCLC 18412261.
• Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
• Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• Schechter, Eric (October 24, 1996). Handbook of Analysis and Its Foundations. Academic Press. ISBN 978-0-08-053299-8.
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

References

1. Swartz 1992, pp. 4–8. - Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. https://search.worldcat.org/oclc/24909067 ↩

2. Assuming that all vector spaces containing a set S {\displaystyle S} are over the same field, when describing the set as being "balanced", it is not necessary to mention a vector space containing S . {\displaystyle S.} That is, " S {\displaystyle S} is balanced" may be written in place of " S {\displaystyle S} is a balanced subset of X {\displaystyle X} ". ↩

3. Narici & Beckenstein 2011, pp. 107–110. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

4. Swartz 1992, pp. 4–8. - Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. https://search.worldcat.org/oclc/24909067 ↩

5. Jarchow 1981, p. 34. - Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. https://search.worldcat.org/oclc/8210342 ↩

6. Narici & Beckenstein 2011, pp. 156–175. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

7. Rudin 1991, pp. 10–14. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

8. Rudin 1991, pp. 10–14. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

9. Let B ⊆ X {\displaystyle B\subseteq X} be balanced. If its topological interior Int X ⁡ B {\displaystyle \operatorname {Int} _{X}B} is empty then it is balanced so assume otherwise and let | s | ≤ 1 {\displaystyle |s|\leq 1} be a scalar. If s ≠ 0 {\displaystyle s\neq 0} then the map X → X {\displaystyle X\to X} defined by x ↦ s x {\displaystyle x\mapsto sx} is a homeomorphism, which implies s Int X ⁡ B = Int X ⁡ ( s B ) ⊆ s B ⊆ B ; {\displaystyle s\operatorname {Int} _{X}B=\operatorname {Int} _{X}(sB)\subseteq sB\subseteq B;} because s Int X ⁡ B {\displaystyle s\operatorname {Int} _{X}B} is open, s Int X ⁡ B ⊆ Int X ⁡ B {\displaystyle s\operatorname {Int} _{X}B\subseteq \operatorname {Int} _{X}B} so that it only remains to show that this is true for s = 0. {\displaystyle s=0.} However, 0 ∈ Int X ⁡ B {\displaystyle 0\in \operatorname {Int} _{X}B} might not be true but when it is true then Int X ⁡ B {\displaystyle \operatorname {Int} _{X}B} will be balanced. ◼ {\displaystyle \blacksquare } /wiki/Homeomorphism ↩

10. Rudin 1991, p. 38. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

11. W {\displaystyle W} being star shaped at the origin means that 0 ∈ W {\displaystyle 0\in W} and r w ∈ W {\displaystyle rw\in W} for all 0 ≤ r ≤ 1 {\displaystyle 0\leq r\leq 1} and w ∈ W . {\displaystyle w\in W.} ↩

12. Rudin 1991, pp. 10–14. - Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi ↩

13. W {\displaystyle W} being star shaped at the origin means that 0 ∈ W {\displaystyle 0\in W} and r w ∈ W {\displaystyle rw\in W} for all 0 ≤ r ≤ 1 {\displaystyle 0\leq r\leq 1} and w ∈ W . {\displaystyle w\in W.} ↩

14. Narici & Beckenstein 2011, pp. 115–154. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

15. Narici & Beckenstein 2011, pp. 107–110. - Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. https://search.worldcat.org/oclc/144216834 ↩

16. Trèves 2006, p. 56. - Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. https://search.worldcat.org/oclc/853623322 ↩

17. Schechter 1996, p. 313. - Schechter, Eric (October 24, 1996). Handbook of Analysis and Its Foundations. Academic Press. ISBN 978-0-08-053299-8. https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA313 ↩

18. Schechter 1996, p. 313. - Schechter, Eric (October 24, 1996). Handbook of Analysis and Its Foundations. Academic Press. ISBN 978-0-08-053299-8. https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA313 ↩