Balanced set

A set $U$ in a real or complex vector space $X$ such that $x \in U$ and $|\lambda| \le 1$ imply $\lambda x \in U$. An example of a balanced set is given by the unit ball in a normed vector space and, generally, by a neighbourhood $U$ of zero in a base of neighbourhoods of zero in a topological vector space. These neighbourhoods of zero are moreover absorbing, i.e. such that for any $x \in X$ there exists an $\alpha > 0$ such that $x \in \lambda U$ for $|\lambda| \ge \alpha$. If $U$ is a convex, absorbing and balanced set, then the functional $p_U(x) = \inf\{|\lambda| : x \in \lambda U\}$ is a semi-norm, i.e. it has the properties $$ p_U(x+y) \le p_U(x) + p_U(y)\ ,\ \ p_U(\lambda x) = |\lambda| p(x) \ . $$

A balanced set is also called centred.

References

[1]	L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)

Comments

The functional $p_U$ mentioned above is also called the Minkowski functional of the convex, absorbing and balanced set $U$.

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5