Seminorm

A seminorm is a function on a vector space V, denoted ||v||, such that the following conditions hold for all v and w in V, and any scalar c.

1. ||v||>=0,

2. ||cv||=|c|||v||, and

3. ||v+w||<=||v||+||w||.

Note that it is possible for ||v||=0 for nonzero v. For example, the functional ||f||=|f(0)| for continuous functions is a seminorm which is not a norm. A seminorm is a norm if ||v||=0 is equivalent to v=0.

See also

This entry contributed by Todd Rowland

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Rowland, Todd. "Seminorm." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Seminorm.html

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