nLab unit ball

Definition

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

One special case that is often used is the unit ball in a metrized topological vector space, whereby the point $x$ 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 $Ban \to Set$ 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.

Related concepts

• disk

• open ball

References

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

• Michael Barr, Heinrich Kleisli, Topological balls, Cahiers Topologie Géométrie Différentielle Catégorique, 40 1 (1999) 3–20 [numdam:CTGDC_1999__40_1_3_0]