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Unit ball

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

\{x\in V: \|x\|<1\}.

The closed unit ball on V under \|\cdot\| is

\{x\in V: \|x\|\le 1\}.

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.

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Science Fair Project Encyclopedia Contents Page
10-26-2009 08:16:03
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