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, $${\displaystyle a}$$ See more

• 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 See more

The empty set is a balanced set. As is any vector subspace of any (real or complex) vector space. In particular, $${\displaystyle \{0\}}$$ is … See more

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 $${\displaystyle B}$$ See more