In mathematics, the open unit ball in a normed vector space V for a given norm is

.

The closed unit ball on V under is

.

The 'shape' of the unit ball is entirely dependent on the chosen norm; it may well have 'corners', and for example may look like [−1,1]ⁿ, in the case of the norm l_∞ in Rⁿ. The round ball is understood as the usual Hilbert space norm, based in the finite dimensional case on the Euclidean distance; its boundary is what is usually meant by the unit sphere.

See also:

• unit circle
• unit square
• unit disk
• sphere
• ball (mathematics)
• Table of mathematical symbols