Balanced set

If C is a convex subset of X, then C is balanced if and only if aC ⊆ C for all scalars a satisfying |a| = 1.[1]

• The open and closed balls centered at 0 in a normed vector space are balanced sets.
• Any subspace of a real or complex vector space is a balanced set.
• The balanced hull of a bounded set is bounded.[2]
• The balanced hull of a totally bounded set is totally bounded.[2]
• 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).
• Consider ${\displaystyle \mathbb {C} ,}$ the field of complex numbers, as a 1-dimensional 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.
• If ${\displaystyle p}$ is a semi-norm on a linear space ${\displaystyle X,}$ then for any constant ${\displaystyle c>0,}$ the set

${\displaystyle \{x\in X\mid p(x)\leqslant c\}}$

is balanced.

• The image of a balanced set under a linear operator is again a balanced set.
• The inverse image of a balanced set (in the codomain) under a linear operator is again a balanced set (in the domain).
• In any topological vector space, the interior of a balanced neighborhood of 0 is again balanced.
• In any topological vector space, the balanced hull of any open neighborhood of 0 is again open.
• The closure of a balanced subset is again balanced.

• The union and intersection of balanced sets is a balanced set.
• The closure of a balanced set is balanced.
• The union of ${\displaystyle \{0\}}$ and the interior of a balanced set is balanced.
• A set is absolutely convex if and only if it is convex and balanced
• If X is a Hausdorff topological vector space and if K is a compact subset of X, then the balanced hull of K is compact.[3]
• If B is balanced then for any scalar s, sB = |s|B.
• If B is balanced then for any scalars a and b such that |a| ≤ |b|, aB ⊆ bB.

• Star domain

• Narici, Lawrence (2011). Topological vector spaces. Boca Raton, FL: CRC Press. ISBN 1-58488-866-0. OCLC 144216834.
• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 4.
• W. Rudin (1990). Functional Analysis (2nd ed.). McGraw-Hill, Inc. ISBN 0-07-054236-8.
• H.H. Schaefer (1970). Topological Vector Spaces. GTM. 3. Springer-Verlag. p. 11. ISBN 0-387-05380-8.
• Trèves, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)