nLab balanced set

Definition

A subset $B$ of a vector space $V$ over $\mathbb{R}$ or $\mathbb{C}$ is balanced if $\lambda x \in B$ whenever $x \in B$ and ${|\lambda|} \le 1$.

Properties

The intersection of two balanced sets is balanced. The closure of a balanced set is again balanced.

Examples

• The unit ball of a seminormed space is balanced
• Any subspace is balanced

Related concepts

• convex set
• absolutely convex set
• norm
• absorbing set