nLab unit ball




In a metric space (X,d)(X,d), the unit ball around some point xXx \in X is the set of points of distance at most 1 from xx: {yX|d(x,y)1}\{y \in X \vert d(x,y) \leq 1\}.

One special case that is often used is the unit ball in a metrized topological vector space, whereby the point xx in the above definition is taken to be the zero vector. More specifically, the unit ball of a Banach space is of interest, as there is an adjunction between the category of Banach spaces and contractions, and the category of sets, where the right adjoint functor BanSetBan \to Set takes the underlying set of the unit ball.

Note that the unit ball around the zero vector makes sense in any normed vector space, or even if the vector space is only equipped with a seminorm.


On star-autonomous categories of unit balls in Banach spaces:

Last revised on October 26, 2023 at 06:43:03. See the history of this page for a list of all contributions to it.