In mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite.
Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
also has the following property:[proof 2] Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties.
Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function.
is called a sublinear function if it is subadditive and positive homogeneous.
Unlike a seminorm, a sublinear function is not necessarily nonnegative.
Sublinear functions are often encountered in the context of the Hahn–Banach theorem.
are intimately tied, via Minkowski functionals, to subsets of
is a sublinear function on a real vector space
[8] Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.
is a sublinear function on a real vector space
into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis at the origin consisting of the following open balls (or the closed balls) centered at the origin:
The resulting topology, pulled back to
about the origin; likewise the closed ball of radius
-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp.
The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms.
are called equivalent if they are both weaker (or both stronger) than each other.
This happens if they satisfy any of the following conditions: A topological vector space (TVS) is said to be a seminormable space (respectively, a normable space) if its topology is induced by a single seminorm (resp.
A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T1 (because a TVS is Hausdorff if and only if it is a T1 space).
A locally bounded topological vector space is a topological vector space that possesses a bounded neighborhood of the origin.
A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.
[17] Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.
[18] A TVS is normable if and only if it is a T1 space and admits a bounded convex neighborhood of the origin.
is a Hausdorff locally convex TVS then the following are equivalent: Furthermore,
is dominated by a positive scalar multiple of
is a continuous seminorm (or more generally, a sublinear function) on
is a linear map between seminormed spaces then the following are equivalent: If
has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.
is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some
A quasi-seminorm that separates points is called a quasi-norm on