Seminormed Space

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.

Definition

Let $X$ be a vector space over either the real numbers $\R$ or the complex numbers $\Complex.$
A real-valued function $p : X \to \R$ is called a if it satisfies the following two conditions:

# Subadditivity/Triangle inequality: $p(x + y) \leq p(x) + p(y)$ for all $x, y \in X.$
# Absolute homogeneity: $p(s x) =, s, p(x)$ for all $x \in X$ and all scalars $s.$

These two conditions imply that $p(0) = 0$If $z \in X$ denotes the zero vector in $X$ while $0$ denote the zero scalar, then absolute homogeneity implies that $p(z) = p(0 z) = , 0, p(z) = 0 p(z) = 0.$ $\blacksquare$ and that every seminorm $p$ also has the following property:Suppose $p : X \to \R$ is a seminorm and let $x \in X.$ Then absolute homogeneity implies $p(-x) = p((-1) x) =, -1, p(x) = p(x).$ The triangle inequality now implies $p(0) = p(x + (- x)) \leq p(x) + p(-x) = p(x) + p(x) = 2 p(x).$ Because $x$ was an arbitrary vector in $X,$ it follows that $p(0) \leq 2 p(0),$ which implies that $0 \leq p(0)$ (by subtracting $p(0)$ from both sides). Thus $0 \leq p(0) \leq 2 p(x)$ which implies $0 \leq p(x)$ (by multiplying thru by $1/2$).

3. Nonnegativity: $p(x) \geq 0$ for all $x \in X.$

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 $X$ is a seminorm that also separates points, meaning that it has the following additional property:

4. Positive definite/: for all $x \in X,$ if $p(x) = 0$ then $x = 0.$

A is a pair $(X, p)$ consisting of a vector space $X$ and a seminorm $p$ on $X.$ If the seminorm $p$ is also a norm then the seminormed space $(X, p)$ is called a .

Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map $p : X \to \R$ 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 $p : X \to \R$ is a seminorm if and only if it is a sublinear and balanced function.

Examples

• The on $X,$ which refers to the constant $0$ map on $X,$ induces the indiscrete topology on $X.$


• If $f$ is any linear form on a vector space then its absolute value $, f, ,$ defined by $x \mapsto , f(x), ,$ is a seminorm.


• A sublinear function $f : X \to \R$ on a real vector space $X$ is a seminorm if and only if it is a , meaning that $f(-x) = f(x)$ for all $x \in X.$


• Every real-valued sublinear function $f : X \to \R$ on a real vector space $X$ induces a seminorm $p : X \to \R$ defined by $p(x) := \max \.$


• 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 $p : X \to \R$ and $q : Y \to \R$ are seminorms (respectively, norms) on $X$ and $Y$ then the map $r : X \times Y \to \R$ defined by $r(x, y) = p(x) + q(y)$ is a seminorm (respectively, a norm) on $X \times Y.$ In particular, the maps on $X \times Y$ defined by $(x, y) \mapsto p(x)$ and $(x, y) \mapsto q(y)$ are both seminorms on $X \times Y.$


• If $p$ and $q$ are seminorms on $X$ then so are
  $(p \vee q)(x) = \max \ \quad \text \quad (p \wedge q)(x) := \inf \$
  where $p \wedge q \leq p$ and $p \wedge q \leq q.$
  


• The space of seminorms on $X$ is generally not a distributive lattice with respect to the above operations. For example, over $\R^2$, $p(x, y) := \max(, x, , , y, ), q(x, y) := 2, x, , r(x, y) := 2, y,$ are such that$((p \vee q) \wedge (p \vee r)) (x, y) = \inf \ \quad \text \quad (p \vee q \wedge r) (x, y) := \max(, x, , , y, )$


• If $L : X \to Y$ is a linear map and $q : Y \to \R$ is a seminorm on $Y,$ then $q \circ L : X \to \R$ is a seminorm on $X.$ The seminorm $q \circ L$ will be a norm on $X$ if and only if $L$ is injective and the restriction $q\big\vert_$ is a norm on $L(X).$

Minkowski functionals and seminorms

Seminorms on a vector space $X$ are intimately tied, via Minkowski functionals, to subsets of $X$ that are convex, balanced, and absorbing. Given such a subset $D$ of $X,$ the Minkowski functional of $D$ is a seminorm. Conversely, given a seminorm $p$ on $X,$ 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 $p.$

Algebraic properties

Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including:
* Convexity
* Reverse triangle inequality: $, p(x) - p(y), \leq p(x - y)$
* For any $r > 0$, $x + \ = \$
* For any $r > 0$, $\$ is an absorbing disk in $X$
* $p(0) = 0$
* $0 \leq \max \$ and $p(x) - p(y) \leq p(x - y)$
* If $p$ is a sublinear function on a real vector space $X$ then there exists a linear functional $f$ on $X$ such that $f \leq p$
* If $X$ is a real vector space, $f$ is a linear functional on $X,$ and $p$ is a sublinear function on $X,$ then $f \leq p$ on $X$ if and only if $f^(1) \cap \$

Other properties of seminorms

Every seminorm is a balanced function.

If p : X \to r \ = \ = \left\.

If $D$ is a set satisfying $\ \subseteq D \subseteq \$ then $D$ is absorbing in $X$ and $p = p_D$ where $p_D$ denotes the Minkowski functional associated with $D$ (that is, the gauge of $D$).
* In particular, if $D$ is as above and $q$ is any seminorm on $X,$ then $q = p$ if and only if $\ \subseteq D \subseteq \.$

• If $(X, \">\,\cdot\,\, )$ is a normed space and $x, y \in X$ then $\, x - y\, = \, x - z\, + \, z - y\,$ for all z \in , y


• 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 $p : X \to \R$ be a non-negative function. The following are equivalent:

1. $p$ is a seminorm.


2. $p$ is a convex $F$-seminorm.


3. $p$ is a convex balanced ''G''-seminorm.

If any of the above conditions hold, then the following are equivalent:

1. $p$ is a norm;


2. $\$ does not contain a non-trivial vector subspace.


3. There exists a norm on $X,$ with respect to which, $\$ is bounded.

If $p$ is a sublinear function on a real vector space $X$ then the following are equivalent:

1. $p$ is a linear functional;


2. $p(x) + p(-x) \leq 0 \text x \in X$;


3. $p(x) + p(-x) = 0 \text x \in X$;

Inequalities involving seminorms

If $p, q : X \to [0, \infty)$ are seminorms on $X$ then:

• $p \leq q$ if and only if $q(x) \leq 1$ implies $p(x) \leq 1.$


• If $a > 0$ and $b > 0$ are such that $p(x) < a$ implies $q(x) \leq b,$ then $a q(x) \leq b p(x)$ for all $x \in X.$


• Suppose $a$ and $b$ are positive real numbers and $q, p_1, \ldots, p_n$ are seminorms on $X$ such that for every $x \in X,$ if $\max \ < a$ then $q(x) < b.$ Then $a q \leq b \left(p_1 + \cdots + p_n\right).$


• If $X$ is a vector space over the reals and $f$ is a non-zero linear functional on $X,$ then $f \leq p$ if and only if $\varnothing = f^(1) \cap \.$

If $p$ is a seminorm on $X$ and $f$ is a linear functional on $X$ then:

• $, f, \leq p$ on $X$ if and only if $\operatorname f \leq p$ on $X$ (see footnote for proof).Obvious if $X$ is a real vector space. For the non-trivial direction, assume that $\operatorname f \leq p$ on $X$ and let $x \in X.$ Let $r \geq 0$ and $t$ be real numbers such that $f(x) = r e^.$ Then $, f(x), = r = f\left(e^ x\right) = \operatorname\left(f\left(e^ x\right)\right) \leq p\left(e^ x\right) = p(x).$


• $f \leq p$ on $X$ if and only if $f^(1) \cap \.$


• If $a > 0$ and $b > 0$ are such that $p(x) < a$ implies $f(x) \neq b,$ then $a , f(x), \leq b p(x)$ for all $x \in X.$

Hahn–Banach theorem for seminorms

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem:
:If $M$ is a vector subspace of a seminormed space $(X, p)$ and if $f$ is a continuous linear functional on $M,$ then $f$ may be extended to a continuous linear functional $F$ on $X$ that has the same norm as $f.$

A similar extension property also holds for seminorms:

:Proof: Let $S$ be the convex hull of $\ \cup \.$ Then $S$ is an absorbing disk in $X$ and so the Minkowski functional $P$ of $S$ is a seminorm on $X.$ This seminorm satisfies $p = P$ on $M$ and $P \leq q$ on $X.$ $\blacksquare$

Topologies of seminormed spaces

Pseudometrics and the induced topology

A seminorm $p$ on $X$ induces a topology, called the , via the canonical translation-invariant pseudometric $d_p : X \times X \to \R$; $d_p(x, y) := p(x - y) = p(y - x).$
This topology is Hausdorff if and only if $d_p$ is a metric, which occurs if and only if $p$ is a norm.
This topology makes $X$ 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:
$\ \quad \text \quad \$
as $r > 0$ ranges over the positive reals.
Every seminormed space $(X, p)$ 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 .

Equivalently, every vector space $X$ with seminorm $p$ induces a vector space quotient $X / W,$ where $W$ is the subspace of $X$ consisting of all vectors $x \in X$ with $p(x) = 0.$ Then $X / W$ carries a norm defined by $p(x + W) = p(v).$ The resulting topology, pulled back to $X,$ is precisely the topology induced by $p.$

Any seminorm-induced topology makes $X$ locally convex, as follows. If $p$ is a seminorm on $X$ and $r \in \R,$ call the set $\$ the ; likewise the closed ball of radius $r$ is $\.$ The set of all open (resp. closed) $p$-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the $p$-topology on $X.$

Stronger, weaker, and equivalent seminorms

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If $p$ and $q$ are seminorms on $X,$ then we say that $q$ is than $p$ and that $p$ is than $q$ if any of the following equivalent conditions holds:

# The topology on $X$ induced by $q$ is finer than the topology induced by $p.$
# If $x_ = \left(x_i\right)_^$ is a sequence in $X,$ then $q\left(x_\right) := \left(q\left(x_i\right)\right)_^ \to 0$ in $\R$ implies $p\left(x_\right) \to 0$ in $\R.$
# If $x_ = \left(x_i\right)_$ is a net in $X,$ then $q\left(x_\right) := \left(q\left(x_i\right)\right)_ \to 0$ in $\R$ implies $p\left(x_\right) \to 0$ in $\R.$
# $p$ is bounded on $\.$
# If $\inf \ = 0$ then $p(x) = 0$ for all $x \in X.$
# There exists a real $K > 0$ such that $p \leq K q$ on $X.$

The seminorms $p$ and $q$ are called if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions:

1. The topology on $X$ induced by $q$ is the same as the topology induced by $p.$


2. $q$ is stronger than $p$ and $p$ is stronger than $q.$


3. If $x_ = \left(x_i\right)_^$ is a sequence in $X$ then $q\left(x_\right) := \left(q\left(x_i\right)\right)_^ \to 0$ if and only if $p\left(x_\right) \to 0.$


4. There exist positive real numbers $r > 0$ and $R > 0$ such that $r q \leq p \leq R q.$

Normability and seminormability

A topological vector space (TVS) is said to be a (respectively, a ) 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 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.
Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.
A TVS is normable if and only if it is a T₁ space and admits a bounded convex neighborhood of the origin.

If $X$ is a Hausdorff locally convex TVS then the following are equivalent:

1. $X$ is normable.


2. $X$ is seminormable.


3. $X$ has a bounded neighborhood of the origin.


4. The strong dual $X^_b$ of $X$ is normable.


5. The strong dual $X^_b$ of $X$ is metrizable.

Furthermore, $X$ is finite dimensional if and only if $X^_$ is normable (here $X^_$ denotes $X^$ 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).

Topological properties

• If $X$ is a TVS and $p$ is a continuous seminorm on $X,$ then the closure of $\$ in $X$ is equal to $\.$


• The closure of $\$ in a locally convex space $X$ whose topology is defined by a family of continuous seminorms $\mathcal$ is equal to $\bigcap_ p^(0).$


• A subset $S$ in a seminormed space $(X, p)$ is bounded if and only if $p(S)$ is bounded.


• If $(X, p)$ is a seminormed space then the locally convex topology that $p$ induces on $X$ makes $X$ into a pseudometrizable TVS with a canonical pseudometric given by $d(x, y) := p(x - y)$ for all $x, y \in X.$


• 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).

Continuity of seminorms

If $p$ is a seminorm on a topological vector space $X,$ then the following are equivalent:

1. $p$ is continuous.


2. $p$ is continuous at 0;


3. $\$ is open in $X$;


4. $\$ is closed neighborhood of 0 in $X$;


5. $p$ is uniformly continuous on $X$;


6. There exists a continuous seminorm $q$ on $X$ such that $p \leq q.$

In particular, if $(X, p)$ is a seminormed space then a seminorm $q$ on $X$ is continuous if and only if $q$ is dominated by a positive scalar multiple of $p.$

If $X$ is a real TVS, $f$ is a linear functional on $X,$ and $p$ is a continuous seminorm (or more generally, a sublinear function) on $X,$ then $f \leq p$ on $X$ implies that $f$ is continuous.

Continuity of linear maps

If $F : (X, p) \to (Y, q)$ is a map between seminormed spaces then let
$\, F\, _ := \sup \.$

If $F : (X, p) \to (Y, q)$ is a linear map between seminormed spaces then the following are equivalent:

1. $F$ is continuous;


2. $\, F\, _ < \infty$;


3. There exists a real $K \geq 0$ such that $p \leq K q$;
   * In this case, $\, F\, _ \leq K.$

If $F$ is continuous then $q(F(x)) \leq \, F\, _ p(x)$ for all $x \in X.$

The space of all continuous linear maps $F : (X, p) \to (Y, q)$ between seminormed spaces is itself a seminormed space under the seminorm $\, F\, _.$
This seminorm is a norm if $q$ is a norm.

Generalizations

The concept of in composition algebras does share the usual properties of a norm.

A composition algebra $(A, *, N)$ consists of an algebra over a field $A,$ an involution $\,*,$ and a quadratic form $N,$ which is called the "norm". In several cases $N$ is an isotropic quadratic form so that $A$ has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article.

An or a is a seminorm $p : X \to \R$ that also satisfies $p(x + y) \leq \max \ \text x, y \in X.$

Weakening subadditivity: Quasi-seminorms

A map $p : X \to \R$ is called a if it is (absolutely) homogeneous and there exists some $b \leq 1$ such that $p(x + y) \leq b p(p(x) + p(y)) \text x, y \in X.$
The smallest value of $b$ for which this holds is called the

A quasi-seminorm that separates points is called a on $X.$

Weakening homogeneity - $k$-seminorms

A map $p : X \to \R$ is called a if it is subadditive and there exists a $k$ such that $0 < k \leq 1$ and for all $x \in X$ and scalars $s,$$p(s x) = , s, ^k p(x)$ A $k$-seminorm that separates points is called a on $X.$

We have the following relationship between quasi-seminorms and $k$-seminorms:

See also

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Notes

Proofs

References

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External links

Sublinear functions

The sandwich theorem for sublinear and super linear functionals

{{DEFAULTSORT:Norm (Mathematics)

Linear algebra