A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.
Definition
Let be a vector space over either the real numbers or the complex numbers
A real-valued function is called a seminorm if it satisfies the following two conditions:
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:
A seminormed space 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 normed space.
Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map 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 is a seminorm if and only if it is a sublinear and balanced function.
Examples
The trivial seminorm 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. if and is the Lebesgue measure) because does not always imply . To make a norm, quotient by the closed subspace of functions with . The resulting space, , has a norm induced by .
A sublinear function on a real vector space is a seminorm if and only if it is a symmetric function, meaning that for all
Every real-valued sublinear function on a real vector space induces a seminorm defined by [2]
Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a vector subspace is once again a seminorm (respectively, norm).
If and are seminorms (respectively, norms) on and then the map defined by is a seminorm (respectively, a norm) on In particular, the maps on defined by and are both seminorms on
If and are seminorms on then so are[3] and
where and [4]
The space of seminorms on is generally not a distributive lattice with respect to the above operations. For example, over , are such that
while
If is a linear map and is a seminorm on then is a seminorm on The seminorm will be a norm on if and only if is injective and the restriction is a norm on
Seminorms on a vector space are intimately tied, via Minkowski functionals, to subsets of that are convex, balanced, and absorbing. Given such a subset of the Minkowski functional of is a seminorm. Conversely, given a seminorm on the sets and are convex, balanced, and absorbing and furthermore, the Minkowski functional of these two sets (as well as of any set lying "in between them") is [5]
For any vector and positive real [7]
and furthermore, is an absorbingdisk in [3]
If is a sublinear function on a real vector space then there exists a linear functional on such that [6] and furthermore, for any linear functional on on if and only if [6]
Other properties of seminorms
Every seminorm is a balanced function.
A seminorm is a norm on if and only if does not contain a non-trivial vector subspace.
If is a seminorm on then is a vector subspace of and for every is constant on the set and equal to [proof 3]
If is a set satisfying then is absorbing in and where denotes the Minkowski functional associated with (that is, the gauge of ).[5] In particular, if is as above and is any seminorm on then if and only if [5]
If is a normed space and then for all in the interval [8]
Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.
Relationship to other norm-like concepts
Let be a non-negative function. The following are equivalent:
Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:
If is a vector subspace of a seminormed space and if is a continuous linear functional on then may be extended to a continuous linear functional on that has the same norm as [15]
A similar extension property also holds for seminorms:
Theorem[16][12](Extending seminorms) — If is a vector subspace of is a seminorm on and is a seminorm on such that then there exists a seminorm on such that and
A seminorm on induces a topology, called the seminorm-induced topology, via the canonical translation-invariantpseudometric;
This topology is Hausdorff if and only if is a metric, which occurs if and only if is a norm.[4]
This topology makes into a locally convexpseudometrizabletopological 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:
as ranges over the positive reals.
Every seminormed space should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called seminormable.
Equivalently, every vector space with seminorm induces a vector space quotient where is the subspace of consisting of all vectors with Then carries a norm defined by The resulting topology, pulled back to is precisely the topology induced by
Any seminorm-induced topology makes locally convex, as follows. If is a seminorm on and call the set the open ball of radius about the origin; likewise the closed ball of radius is The set of all open (resp. closed) -balls at the origin forms a neighborhood basis of convexbalanced sets that are open (resp. closed) in the -topology on
Stronger, weaker, and equivalent seminorms
The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If and are seminorms on then we say that is stronger than and that is weaker than if any of the following equivalent conditions holds:
The topology on induced by is finer than the topology induced by
The seminorms and are called equivalent if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions:
The topology on induced by is the same as the topology induced by
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 single norm).
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.
Normability of topological vector spaces is characterized by Kolmogorov's normability criterion.
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.
If is a Hausdorff locally convex TVS then the following are equivalent:
Furthermore, is finite dimensional if and only if is normable (here denotes endowed with the weak-* topology).
The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional).[18]
Topological properties
If is a TVS and is a continuous seminorm on then the closure of in is equal to [3]
The closure of in a locally convex space whose topology is defined by a family of continuous seminorms is equal to [11]
A subset in a seminormed space is bounded if and only if is bounded.[20]
If is a seminormed space then the locally convex topology that induces on makes into a pseudometrizable TVS with a canonical pseudometric given by for all [21]
The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces are trivial (that is, 0-dimensional).[18]
Continuity of seminorms
If is a seminorm on a topological vector space then the following are equivalent:[5]
There exists a continuous seminorm on such that [3]
In particular, if is a seminormed space then a seminorm on is continuous if and only if is dominated by a positive scalar multiple of [3]
If is a real TVS, is a linear functional on and is a continuous seminorm (or more generally, a sublinear function) on then on implies that is continuous.[6]
Continuity of linear maps
If is a map between seminormed spaces then let[15]
If is a linear map between seminormed spaces then the following are equivalent:
The space of all continuous linear maps between seminormed spaces is itself a seminormed space under the seminorm
This seminorm is a norm if is a norm.[15]
Generalizations
The concept of norm in composition algebras does not share the usual properties of a norm.
An ultraseminorm or a non-Archimedean seminorm is a seminorm that also satisfies
Weakening subadditivity: Quasi-seminorms
A map is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some such that
The smallest value of for which this holds is called the multiplier of
A quasi-seminorm that separates points is called a quasi-norm on
Weakening homogeneity - -seminorms
A map is called a -seminorm if it is subadditive and there exists a such that and for all and scalars A -seminorm that separates points is called a -norm on
We have the following relationship between quasi-seminorms and -seminorms:
Suppose that is a quasi-seminorm on a vector space with multiplier If then there exists -seminorm on equivalent to
^If denotes the zero vector in while denote the zero scalar, then absolute homogeneity implies that
^Suppose is a seminorm and let Then absolute homogeneity implies The triangle inequality now implies Because was an arbitrary vector in it follows that which implies that (by subtracting from both sides). Thus which implies (by multiplying thru by ).
^Let and It remains to show that The triangle inequality implies Since as desired.
Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN978-3-540-08662-8. OCLC297140003.
Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN978-0-387-90081-0. OCLC878109401.
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN978-3-642-64988-2. MR0248498. OCLC840293704.
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