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Concept
Seminorm
Summary
In mathematics, particularly in functional analysis, a seminorm is a vector space norm that 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.
A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.
Let be a vector space over either the real numbers or the complex numbers
A real-valued function is called a if it satisfies the following two conditions:
Subadditivity/Triangle inequality: for all
Absolute homogeneity: for all and all scalars
These two conditions imply that and that every seminorm also has the following property:
Nonnegativity: for all
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.
By definition, a norm on is a seminorm that also separates points, meaning that it has the following additional property:
Positive definite/Positive/: whenever satisfies then
A is a pair consisting of a vector space and a seminorm on If the seminorm is also a norm then the seminormed space is called a .
Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map is called a if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem.
A real-valued function is a seminorm if and only if it is a sublinear and balanced function.
The on which refers to the constant map on induces the indiscrete topology on
Let be a measure on a space . For an arbitrary constant , let be the set of all functions for which
exists and is finite. It can be shown that is a vector space, and the functional is a seminorm on . However, it is not always a norm (e.g.
Official source
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