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In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space is a set S {\displaystyle S} such that a S ⊆ S {\displaystyle aS\subseteq S} for all scalars a {\displaystyle a} satisfying | a | ≤ 1. {\displaystyle |a|\leq 1.}
The balanced hull or balanced envelope of a set S {\displaystyle S} is the smallest balanced set containing S . {\displaystyle S.} The balanced core of a subset S {\displaystyle S} is the largest balanced set contained in S . {\displaystyle S.}
Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin. This neighborhood can also be chosen to be an open set or, alternatively, a closed set.