Seminorm

In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of "length". A space with such a seminorm is then known as a seminormed space. The distinction between a seminorm and a norm (and hence between a seminormed space and a normed space) is that a seminorm may assign zero length to nonzero vectors. Consequently, every norm is a seminorm, and a seminorm is a norm precisely when the only vector measured as 0 is the zero vector. In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm.

More formally, a seminorm on a real or complex vector space V is a function p from V to the non-negative real numbers satisfying

1. Scaling property: p( a x ) = |a| p(x)
2. Subadditive property: p(x+y) ≤ p(x) + p(y)

The pair (V,p) is then a seminormed space.

Apart from normed spaces, the simplest examples of seminormed spaces are the trivial seminorms -- those where p(x) = 0 for all x in V. Product spaces where one of the factors has trivial seminorm, such as R² with p(x,y) = |x| furnish further finite-dimensional examples. Moreover, a straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminorms occur for infinite-dimensional vector spaces.

Seminormed spaces arise in mainstream functional analysis in many situations, in particular,

• The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero. For instance, with the L^p spaces, the function defined by

<math>\|f\|_p = \left( \int |f(x)|^p \;dx \right)^{1/p}<math>

is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.

• A characterisation of Locally convex topological vector spaces has the topology defined by a family of seminorms, and many examples use this characterisation in their definitions. Typically this family is infinite, and there are enough seminorms to distinguish between elements of the vector space, creating a Hausdorff space.