@norms 8

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.
A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.
The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".
A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "




≤



{\displaystyle \,\leq \,}
" in the homogeneity axiom.
It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set.

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