Balanced set

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

Notation

If ${\displaystyle S}$ is a set, ${\displaystyle a}$ is a scalar, and ${\displaystyle B\subseteq \mathbb {K} }$ then let ${\displaystyle aS=\{as:s\in S\}}$ and ${\displaystyle BS=\{bs:b\in B,s\in S\}}$ and for any ${\displaystyle 0\leq r\leq \infty ,}$ let

$${\displaystyle B_{r}=\{a\in \mathbb {K}$$ :|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 ${\displaystyle r}$ in the scalar field ${\displaystyle \mathbb {K} }$ centered at ${\displaystyle 0}$ where ${\displaystyle B_{0}=\varnothing ,B_{\leq 0}=\{0\},}$ and ${\displaystyle B_{\infty }=B_{\leq \infty }=\mathbb {K} .}$ Every balanced subset of the field ${\displaystyle \mathbb {K} }$ is of the form ${\displaystyle B_{\leq r}}$ or ${\displaystyle B_{r}}$ for some ${\displaystyle 0\leq r\leq \infty .}$

Balanced set

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

1. Definition: ${\displaystyle as\in S}$ for all ${\displaystyle s\in S}$ and all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|\leq 1.}$
2. ${\displaystyle aS\subseteq S}$ for all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|\leq 1.}$
3. ${\displaystyle B_{\leq 1}S\subseteq S,}$ where ${\displaystyle B_{\leq 1}:=\{a\in \mathbb {K}$ :|a|\leq 1\}.}
4. ${\displaystyle S=B_{\leq 1}S.}$[1]
5. For every ${\displaystyle s\in S,}$ ${\displaystyle S\cap \mathbb {K} s=B_{\leq 1}(S\cap \mathbb {K} s).}$
   • ${\displaystyle \mathbb {K} s=\operatorname {span} \{s\}}$ is a ${\displaystyle 0}$ (if ${\displaystyle s=0}$) or ${\displaystyle 1}$ (if ${\displaystyle s\neq 0}$) dimensional vector subspace of ${\displaystyle X.}$
   • If ${\displaystyle R:=S\cap \mathbb {K} s}$ then the above equality becomes ${\displaystyle R=B_{\leq 1}R,}$ which is exactly the previous condition for a set to be balanced. Thus, ${\displaystyle S}$ is balanced if and only if for every ${\displaystyle s\in S,}$ ${\displaystyle A\cap \mathbb {K} s}$ is a balanced set (according to any of the previous defining conditions).
6. For every 1-dimensional vector subspace ${\displaystyle Y}$ of ${\displaystyle \operatorname {span} S,}$ ${\displaystyle S\cap Y}$ is a balanced set (according to any defining condition other than this one).
7. For every ${\displaystyle s\in S,}$ there exists some ${\displaystyle 0\leq r\leq \infty }$ such that ${\displaystyle S\cap \mathbb {K} s=B_{r}s}$ or ${\displaystyle S\cap \mathbb {K} s=B_{\leq r}s.}$

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

8. ${\displaystyle aS\subseteq S}$ for all scalars ${\displaystyle a}$ satisfying ${\displaystyle |a|=1.}$[2]

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

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

Balanced hull

$${\displaystyle \operatorname {bal} S=\bigcup _{|a|\leq 1}aS=B_{\leq 1}S}$$

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

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

Balanced core

$${\displaystyle \operatorname {balcore} ={\begin{cases}\bigcap _{|a|\geq 1}aS&{\text{ if }}0\in S\\\varnothing &{\text{ if }}0\not \in S\\\end{cases}}}$$

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

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

• The empty set is a balanced set.
• Any vector subspace of a real or complex vector space is a balanced set.
• The open and closed balls centered at the origin in a normed vector space are balanced sets. If ${\displaystyle p}$ is a seminorm (or norm) on a vector space ${\displaystyle X}$ then for any constant ${\displaystyle c>0,}$ the set ${\displaystyle \{x\in X:p(x)\leq c\}}$ is balanced.
• If ${\displaystyle S\subseteq X}$ is any subset and ${\displaystyle B_{1}:=\{a\in \mathbb {K}$ :|a|<1\}} then ${\displaystyle B_{1}S}$ is a balanced set.
  • In particular, if ${\displaystyle U\subseteq X}$ is any balanced neighborhood of the origin in a topological vector space ${\displaystyle X}$ then ${\displaystyle \operatorname {Int} _{X}U\subseteq B_{1}U=\bigcup _{0<|a|<1}aU\subseteq U.}$
• If ${\displaystyle \mathbb {K} }$ is the field real or complex numbers and ${\displaystyle X=\mathbb {K} }$ is the normed space over ${\displaystyle \mathbb {K} }$ with the usual Euclidean norm, then the balanced subsets of ${\displaystyle X}$ are exactly the following:[3]
  1. ${\displaystyle \varnothing }$
  2. ${\displaystyle X}$
  3. ${\displaystyle \{0\}}$
  4. ${\displaystyle \{x\in X:|x|<r\}}$ for some real ${\displaystyle r>0}$
  5. ${\displaystyle \{x\in X:|x|\leq r\}}$ for some real ${\displaystyle r>0.}$
• If ${\displaystyle X=\mathbb {R} ^{2}}$ (${\displaystyle X}$ is a vector space over ${\displaystyle \mathbb {R} }$), ${\displaystyle B_{\leq 1}}$ is the closed unit ball in ${\displaystyle X}$ centered at the origin, ${\displaystyle x_{0}\in X}$ is non-zero, and ${\displaystyle L:=\mathbb {R} x_{0},}$ then the set ${\displaystyle R:=B_{\leq 1}\cup L}$ is a closed, symmetric, and balanced neighborhood of the origin in ${\displaystyle X.}$ More generally, if ${\displaystyle C}$ is any closed subset of ${\displaystyle X}$ such that ${\displaystyle (0,1)C\subseteq C,}$ then ${\displaystyle S:=B_{\leq 1}\cup C\cup (-C)}$ is a closed, symmetric, and balanced neighborhood of the origin in ${\displaystyle X.}$ This example can be generalized to ${\displaystyle \mathbb {R} ^{n}}$ for any integer ${\displaystyle n\geq 1.}$
• The Cartesian product of a family of balanced sets is balanced in the product space of the corresponding vector spaces (over the same field ${\displaystyle \mathbb {K} }$).
• Consider ${\displaystyle \mathbb {C} ,}$ the field of complex numbers, as a 1-dimensional complex vector space. The balanced sets are ${\displaystyle \mathbb {C} }$ 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, ${\displaystyle \mathbb {C} }$ and ${\displaystyle \mathbb {R} ^{2}}$ are entirely different as far as scalar multiplication is concerned.
• Let ${\displaystyle X=\mathbb {R} ^{2}}$ and let ${\displaystyle B}$ be the union of the line segment between ${\displaystyle (-1,0)}$ and ${\displaystyle (1,0)}$ and the line segment between ${\displaystyle (0,-1)}$ and ${\displaystyle (0,1).}$ Then ${\displaystyle B}$ is balanced but not convex or absorbing. However, ${\displaystyle \operatorname {span} B=X.}$
• Let ${\displaystyle X=\mathbb {R} ^{2}}$ and for every ${\displaystyle 0\leq t\leq \pi ,}$ let ${\displaystyle r_{t}}$ be any positive real number and let ${\displaystyle B^{t}}$ be the (open or closed) line segment between the points ${\displaystyle (\cos t,\sin t)}$ and ${\displaystyle -(\cos t,\sin t).}$ Then the set ${\displaystyle B=\bigcup _{0\leq t<\pi }r_{t}B^{t}}$ is balanced and absorbing but it is not necessarily convex.
• The balanced hull of a closed set need not be closed. Take for instance the graph of ${\displaystyle xy=1}$ in${\displaystyle X=\mathbb {R} ^{2}.}$
• This 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 ${\displaystyle X:=\mathbb {R} ^{2}}$ and let the convex subset be ${\displaystyle S:=[-1,1]\times \{1\},}$ which is a horizontal closed line segment lying above the ${\displaystyle x-}$axis. The balanced hull ${\displaystyle \operatorname {bal} S}$ is a non-convex subset that is "hour glass shaped" and equal to the union of two closed and filled isosceles triangles ${\displaystyle T_{1}}$ and ${\displaystyle T_{2},}$ where ${\displaystyle T_{2}=-T_{1}}$ and ${\displaystyle T_{1}}$ is the filled triangle whose vertices are the origin together with the endpoints of ${\displaystyle S}$ (said differently, ${\displaystyle T_{1}}$ is the convex hull of ${\displaystyle S\cup \{(0,0)\}}$ while ${\displaystyle T_{2}}$ is the convex hull of ${\displaystyle (-S)\cup \{(0,0)\}}$).

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.

• If ${\displaystyle B}$ is balanced then for any scalars ${\displaystyle a}$ and ${\displaystyle b}$ such that ${\displaystyle |a|\leq |b|,}$ ${\displaystyle aB\subseteq bB}$ and ${\displaystyle aB=|a|B.}$
• If ${\displaystyle B}$ is a balanced subset of ${\displaystyle X,}$ then ${\displaystyle B}$ is absorbing in ${\displaystyle X}$ if and only if for all ${\displaystyle x\in X,}$ there exists ${\displaystyle r>0}$ such that ${\displaystyle x\in rB.}$[2]
• If ${\displaystyle B}$ is a balanced subset of ${\displaystyle X}$ then ${\displaystyle B}$ is absorbing in ${\displaystyle \operatorname {span} B.}$
• Every balanced set is star-shaped (at 0) and a symmetric set.
• Suppose ${\displaystyle B}$ is balanced. If ${\displaystyle Y}$ is a 1-dimensional vector subspace of ${\displaystyle X}$ then ${\displaystyle B\cap Y}$ is convex and balanced. If ${\displaystyle Y}$ is a 1-dimensional vector subspace of ${\displaystyle \operatorname {span} B}$ then ${\displaystyle B\cap Y}$ is also absorbing in ${\displaystyle Y.}$
• Suppose ${\displaystyle B\neq \varnothing }$ is a balanced subset of ${\displaystyle X}$ and ${\displaystyle x\in X.}$ Then ${\displaystyle B\cap \operatorname {span} x}$ is a convex balanced neighborhood of ${\displaystyle 0}$ in ${\displaystyle \operatorname {span} x}$ when this 0 or 1-dimensional vector space is endowed with the Hausdorff Euclidean topology. The set ${\displaystyle B\cap \mathbb {R} x}$ is a convex balanced subset of ithe real vector space ${\displaystyle \mathbb {R} x}$ that contains the origin.

Properties of balanced hulls

• ${\displaystyle a\operatorname {bal} S=\operatorname {bal} (aS)}$ for any subset ${\displaystyle S}$ of ${\displaystyle X}$ and any scalar ${\displaystyle a.}$
• ${\displaystyle \operatorname {bal} \left(\bigcup _{S\in {\mathcal {S}}}S\right)=\bigcup _{S\in {\mathcal {S}}}\operatorname {bal} S}$ for any collection ${\displaystyle {\mathcal {S}}}$ of subsets of ${\displaystyle X.}$
• In any topological vector space, the balanced hull of any open neighborhood of the origin is again open.
• If ${\displaystyle X}$ is a Hausdorff topological vector space and if ${\displaystyle K}$ is a compact subset of ${\displaystyle X}$ then the balanced hull of ${\displaystyle K}$ is compact.[7]

Properties of balanced cores

If a set is closed (respectively, convex, absorbing, a neighborhood of the origin) then the same is true of its balanced core.

• Absolutely convex 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 that intersects every line into a single line segment
• Star domain
• Symmetric set
• Topological vector space – Vector space with a notion of nearness

Proofs

• 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.
• 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.