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. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.
Definition Let X {\displaystyle X} be a vector space over either the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C . {\displaystyle \mathbb {C} .} A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } is called a seminorm if it satisfies the following two conditions:
Subadditivity/Triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)} for all x , y ∈ X . {\displaystyle x,y\in X.}
Absolute homogeneity: p ( s x ) = | s | p ( x ) {\displaystyle p(sx)=|s|p(x)} for all x ∈ X {\displaystyle x\in X} and all scalars s . {\displaystyle s.}
These two conditions imply that p ( 0 ) = 0 {\displaystyle p(0)=0} and that every seminorm p {\displaystyle p} also has the following property:
Nonnegativity: p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all x ∈ X . {\displaystyle x\in X.}
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 X {\displaystyle X} is a seminorm that also separates points, meaning that it has the following additional property:
Positive definite/Positive/Point-separating: whenever x ∈ X {\displaystyle x\in X} satisfies p ( x ) = 0 , {\displaystyle p(x)=0,} then x = 0. {\displaystyle x=0.}
A seminormed space is a pair ( X , p ) {\displaystyle (X,p)} consisting of a vector space X {\displaystyle X} and a seminorm p {\displaystyle p} on X . {\displaystyle X.} If the seminorm p {\displaystyle p} is also a norm then the seminormed space ( X , p ) {\displaystyle (X,p)} is called a normed space. Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map p : X → R {\displaystyle p:X\to \mathbb {R} } 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. A real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } is a seminorm if and only if it is a sublinear and balanced function.
Examples
The trivial seminorm on X , {\displaystyle X,} which refers to the constant 0 {\displaystyle 0} map on X , {\displaystyle X,} induces the indiscrete topology on X . {\displaystyle X.}
Let μ {\displaystyle \mu } be a measure on a space Ω {\displaystyle \Omega } . For an arbitrary constant c ≥ 1 {\displaystyle c\geq 1} , let X {\displaystyle X} be the set of all functions f : Ω → R {\displaystyle f:\Omega \rightarrow \mathbb {R} } for which
‖ f ‖ c := ( ∫ Ω | f | c d μ ) 1 / c {\displaystyle \lVert f\rVert _{c}:=\left(\int _{\Omega }|f|^{c}\,d\mu \right)^{1/c}}
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