Seminorm


In mathematics, particularly in functional analysis, a seminorm is a real-valued function on a vector space over the field of real or complex numbers that generalizes the notion of a norm by removing the positive definite requirement of a norm.
This means that unlike a norm, a seminorm is allowed to potentially map non-zero vectors to zero.
A seminormed space is a pair consisting of a vector space and a seminorm on it.
Every norm is a seminorm and every normed space is a seminormed space, but there are seminorms that are not norms.
Seminorms are intimately connected with convex sets and locally convex topological vector spaces.
In fact, a topological vector space is locally convex if and only if its topology is induced by a family of seminorms.

Definition

Throughout, will be a vector space over the field, where is either the real numbers or the complex numbers.
A scalar is any element in 's field.

Seminorms

Definition: A map is called a seminorm if it satisfies the following two conditions:

  1. Subadditivity: for all ;
  2. Absolute homogeneity: for all and all scalars ;
Definition: A seminorm is called a norm if in addition it satisfies the following condition:

  1. Positive definiteness/Separates points: If is such that, then ;
Every seminorm has the following properties:

  1. Nonnegativity: .
    • Note that this condition is also often called positivity even though the requirement is still
  2. Positive homogeneity: for all and all positive real ;

Sublinear functions

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function.
Thus, any result that is known about sublinear functions can be immediately applied to seminorms.
Definition: A map is called a sublinear function if it is subadditive and positively homogeneous.
Unlike a seminorm, a sublinear function is not necessarily nonnegative.
Sublinear functions are often encountered in the context of the Hahn-Banach theorem.

Other related functions

;Ultraseminorm
Definition: A map is called an ultraseminorm or a non-Archimedean seminorm if it is a seminorm and satisfies the following additional condition:

  1. for all.
;Quasi-seminorm
Definition: A map is called a quasi-seminorm if it is absolutely homogeneous and satisfies the following additional condition:

  1. There exists some such that for all,
where 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.
Note that the definition of a quasi-seminorm is the almost the same as the definition of a seminorm, except that subadditivity is replaced by a weaker condition.
;-seminorms
Definition: A map is called a -seminorm if it is subadditive and satisfies the following additional condition:

  1. there exists a such that and for all and scalars :
A -seminorm that separates points is called a -norm on.
Note that the definition of a quasi-seminorm is the almost the same as the definition of a seminorm, except that absolute homogeneity is replaced by a weaker condition.
We have the following relationship between quasi-seminorms and -seminorms:

Induced topology and pseudometric

The canonical pseudometric is a metric if and only if is a norm.
The -topology on induced by a seminorm makes into a locally convex pseudometrizable topological vector space.
This topology is Hausdorff if and only if is a norm.
This topology make into a complete TVS if and only if the canonical pseudometric is a complete pseudometric.
The set of all open -balls at the origin forms a neighborhood basis of convex balanced sets that are open in the -topology on.

Stronger, weaker, and equivalent seminorms


  1. The topology on induced by is finer than the topology induced by.
  2. There exists a real such that on.
  3. is bounded on

    Seminormed spaces

    If is a linear map between seminormed spaces then the following are equivalent:

    1. is continuous;
    2. ;
    3. There exists a real such that ;
      • In this case,.
    If is continuous then for all.
    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.

    Characterizations

    A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.
    A locally convex TVS is seminormable if and only if it has a non-empty bounded open set.
    ;When seminorms are norms
    If is a seminorm on then the following are equivalent:

    1. is a norm;
    2. for all, if and only if ;
    3. does not contain a non-trivial vector subspace.
    4. there exists a Hausdorff TVS topology on in which is bounded.
    ;When sublinear functions are linear
    If is a sublinear function on a real vector space then the following are equivalent:

    1. is a linear functional;
    2. for every, ;
    3. for every, ;

    Examples


    • All norms are seminorms and all seminorms are sublinear functions. The converse, however, is not true in general.
    • Every vector space admits a norm: If is a Hamel basis for a vector space then the real-valued map that sends to is a paranorm on is a norm on.
    • If and are seminorms on then so is.
    • If and are seminorms on then so is. Moreover, and.
    • Any finite sum of seminorms is a seminorm.
    • If is a linear functional on then the map

      Minkowski functionals and seminorms

      Seminorms on a vector space are intimately tied, via Minkowski functionals, to subsets of that are convex, balanced, and absorbing in the following way.
      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 is equal to.
      Thus, seminorms on can be identified with subsets of.
      This is beneficial because there are things that we can do with subsets of that we cannot do with seminorms.

      When Minkowski functionals are seminorms

      Let be a vector space and let be an absorbing disk in .

      Relationship with seminorms

      If is a seminorm on and is a set satisfying
      then is absorbing in and, where is the Minkowski functional associated with .
      • In particular, if is as above and is any seminorm on, then if and only if.

        Properties of Minkowski functionals

      Let be a real or complex vector space and let be an absorbing disk in.

      1. is a seminorm on.
      2. For any scalar,.
      3. If is an absorbing disk in and then
      4. is a norm on if and only if does not contain a non-trivial vector subspace.

      Poperties

      Algebraic properties

      Let be a vector space over where is either the real or complex numbers.
      ;Properties of sublinear functions
      Since every seminorm is a sublinear function, seminorms have all of the following properties:
      If be a real-valued sublinear function on then:

      • Every sublinear function is a convex functional.
      • .
      • and for all.
      • If is a sublinear function on a real vector space then there exists a linear functional on such that.
      • If is a real vector space, is a linear functional on, and is a sublinear function on, then on if and only if.
      ;Properties of seminorms
      If is a seminorm on then:

      • The second triangle inequality: for all.
      • For all, is an absorbing disk in.
      • is a norm on if and only if does not contain a non-trivial vector subspace.
      • Every seminorm is a non-negative sublinear function. However, sublinear functions are not necessarily non-negative.
      • is a vector subspace of.
      • For any and,
      • For any,
      • If is a set satisfying then is absorbing in and, where is the Minkowski functional associated with .
        • In particular, if is as above and is any seminorm on, then if and only if.
      Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable.
      ;Inequalities involving seminorms
      If is a seminorm on then:

      • If is a linear functional on a real or complex vector space and if is a seminorm on, then on if and only if on .
      • If is a seminorm on, then if and only if implies.
      • If is a seminorm on and and are such that implies, then for all.
      • If is a linear functional on and and are such that implies, then for all.
      • If is a vector space over the reals and is a non-0 linear functional on, then if and only if.
      • Suppose and are positive real numbers and are seminorms on. If for every, implies for all, then.

      Topological properties


      • If is a TVS and is a continuous seminorm on, then the closure of in is equal to.
      • The closure of in a locally convex space whose topology is defined by a family of continuous seminorms is equal to.
      • A subset in a seminormed space is bounded if and only if is bounded.
      ;Pseudometrizability

      Continuity

      ;Continuity and sublinear functions
      Suppose is a sublinear function on a TVS.
      Then the following are equivalent:

      1. is continuous;
      2. is continuous at 0;
      3. is uniformly continuous on ;
      and if is non-negative then we may add to this list:

      1. is open in.
      If is a real TVS, is a linear functional on, and is a continuous sublinear function on, then on implies that is continuous.
      ;Continuity of seminorms
      If is a seminorm on a topological vector space, then the following are equivalent:

      1. is continuous.
      2. is continuous at 0;
      3. is open in ;
      4. is closed neighborhood of 0 in ;
      5. is uniformly continuous on ;
      6. There exists a continuous seminorm on such that.
      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.

      Normability

      A topological vector space is called normable if the topology of the space can be induced by a norm.
      Normability of topological vector spaces is characterized by Kolmogorov's normability criterion.
      If is a Hausdorff locally convex TVS then the following are equivalent:

      1. is normable.
      2. has a bounded neighborhood of the origin.
      3. the strong dual of is normable.
      4. the strong dual of is metrizable.
      Furthermore, is finite dimensional if and only if is normable.
      The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial.

      Generalizations

      The concept of norm in composition algebras does not share the usual properties of a norm.
      A composition algebra consists of an algebra over a field, an involution *, and a quadratic form, which is called the "norm".
      In several cases is an isotropic quadratic form so that has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.