Seminorm

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

1. Subadditivity/Triangle inequality: $p(x+y)\le p(x)+p(y)$ for all $x,y\in X.$
2. Absolute homogeneity: $p(sx)=|s|p(x)$ for all $x\in X$ and all scalars $s.$

These two conditions imply that $p(0)=0$ and that every seminorm $p$ also has the following property:

3. Nonnegativity: $p(x)\ge 0$ 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/Positive/Point-separating: whenever $x\in X$ satisfies $p(x)=0,$ then $x=0.$

A seminormed space 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 normed space.

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 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 $p:X\to ℝ$ is a seminorm if and only if it is a sublinear and balanced function.

Examples

• The trivial seminorm on $X,$ which refers to the constant $0$ map on $X,$ induces the indiscrete topology on $X.$
• Let $\mu$ be a measure on a space $\Omega$. For an arbitrary constant $c\ge 1$, let $X$ be the set of all functions $f:\Omega \to ℝ$ for which $$\Vert f{\Vert }_c:={({\int }_{\Omega }|f{|}^{c}d\mu )}^{1/c}$$ exists and is finite. It can be shown that $X$ is a vector space, and the functional $\Vert \cdot {\Vert }_c$ is a seminorm on $X$. However, it is not always a norm (e.g. if $\Omega =ℝ$ and $\mu$ is the Lebesgue measure) because $\Vert h{\Vert }_c=0$ does not always imply $h=0$. To make $\Vert \cdot {\Vert }_c$ a norm, quotient $X$ by the closed subspace of functions $h$ with $\Vert h{\Vert }_c=0$. The resulting space, $L^c(\mu )$, has a norm induced by $\Vert \cdot {\Vert }_c$.
• 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 symmetric function, 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 $p(x):=max\{f(x),f(-x)\}.$
• 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 $$(p\vee q)(x)=max\{p(x),q(x)\}$$ and $$(p\wedge q)(x):=inf\{p(y)+q(z):x=y+z\text{ with }y,z\in X\}$$ where $p\wedge q\le p$ and $p\wedge q\le q.$
• 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 $$((p\vee q)\wedge (p\vee r))(x,y)=inf\{max(2|x_1|,|y_1|)+max(|x_2|,2|y_2|):x=x_1+x_2\text{ and }y=y_1+y_2\}$$ while $(p\vee q\wedge r)(x,y):=max(|x|,|y|)$
• 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 $q{|}_{L(X)}$ 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$\{x\in X:p(x)<1\}$ and $\{x\in X:p(x)\le 1\}$ 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, $p(0)=0,$ and for all vectors $x,y\in X$: the reverse triangle inequality: $$|p(x)-p(y)|\le p(x-y)$$ and also $0\le max\{p(x),p(-x)\}$ and $p(x)-p(y)\le p(x-y).$

For any vector $x\in X$ and positive real $r>0:$ $$x+\{y\in X:p(y)<r\}=\{y\in X:p(x-y)<r\}$$ and furthermore, $\{x\in X:p(x)<r\}$ is an absorbing disk in $X.$

If $p$ is a sublinear function on a real vector space $X$ then there exists a linear functional $f$ on $X$ such that $f\le p$ and furthermore, for any linear functional $g$ on $X,$ $g\le p$ on $X$ if and only if $g^{-1}(1)\cap \{x\in X:p(x)<1\}=\varnothing .$

Other properties of seminorms

Every seminorm is a balanced function. A seminorm $p$ is a norm on $X$ if and only if $\{x\in X:p(x)<1\}$ does not contain a non-trivial vector subspace.

If $p:X\to [0,\mathrm{\infty })$ is a seminorm on $X$ then $\ker p:=p^{-1}(0)$ is a vector subspace of $X$ and for every $x\in X,$ $p$ is constant on the set $x+\ker p=\{x+k:p(k)=0\}$ and equal to $p(x).$

Furthermore, for any real $r>0,$ $$r\{x\in X:p(x)<1\}=\{x\in X:p(x)<r\}=\{x\in X:\frac{1}{r}p(x)<1\}.$$

If $D$ is a set satisfying $\{x\in X:p(x)<1\}\subseteq D\subseteq \{x\in X:p(x)\le 1\}$ 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 $\{x\in X:q(x)<1\}\subseteq D\subseteq \{x\in X:q(x)\le \}.$

If $(X,\Vert \cdot \Vert )$ is a normed space and $x,y\in X$ then $\Vert x-y\Vert =\Vert x-z\Vert +\Vert z-y\Vert$ for all $z$ in the interval $[x,y].$

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. $\{x\in X:p(x)<1\}$ does not contain a non-trivial vector subspace.
3. There exists a norm on $X,$ with respect to which, $\{x\in X:p(x)<1\}$ 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. $p(x)+p(-x)\le 0\text{ for every }x\in X$;
3. $p(x)+p(-x)=0\text{ for every }x\in X$;

Inequalities involving seminorms

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

• $p\le q$ if and only if $q(x)\le 1$ implies $p(x)\le 1.$
• If $a>0$ and $b>0$ are such that $p(x)<a$ implies $q(x)\le b,$ then $aq(x)\le bp(x)$ for all $x\in X.$
• Suppose $a$ and $b$ are positive real numbers and $q,p_1,\dots ,p_n$ are seminorms on $X$ such that for every $x\in X,$ if $max\{p_1(x),\dots ,p_n(x)\}<a$ then $q(x)<b.$ Then $aq\le b(p_1+\cdots +p_n).$
• If $X$ is a vector space over the reals and $f$ is a non-zero linear functional on $X,$ then $f\le p$ if and only if $\varnothing =f^{-1}(1)\cap \{x\in X:p(x)<1\}.$

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

• $|f|\le p$ on $X$ if and only if $\mathrm{Re}f\le p$ on $X$ (see footnote for proof).
• $f\le p$ on $X$ if and only if $f^{-1}(1)\cap \{x\in X:p(x)<1=\varnothing \}.$
• If $a>0$ and $b>0$ are such that $p(x)<a$ implies $f(x)\ne b,$ then $a|f(x)|\le bp(x)$ 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:

Theorem (Extending seminorms)—If $M$ is a vector subspace of $X,$ $p$ is a seminorm on $M,$ and $q$ is a seminorm on $X$ such that $p\le q{|}_M,$ then there exists a seminorm $P$ on $X$ such that $P{|}_M=p$ and $P\le q.$

Proof: Let $S$ be the convex hull of $\{m\in M:p(m)\le 1\}\cup \{x\in X:q(x)\le 1\}.$ 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 $P\le q$ on $X.$ $◼$

Topologies of seminormed spaces

Pseudometrics and the induced topology

A seminorm $p$ on $X$ induces a topology, called the seminorm-induced topology, 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: $$\{x\in X:p(x)<r\}\text{ or }\{x\in X:p(x)\le r\}$$ 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 seminormable.

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 $p(x+W)=p(x).$ 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 $\{x\in X:p(x)<r\}$ the open ball of radius $r$ about the origin; likewise the closed ball of radius $r$ is $\{x\in X:p(x)\le r\}.$ 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 stronger than $p$ and that $p$ is weaker than $q$ if any of the following equivalent conditions holds:

1. The topology on $X$ induced by $q$ is finer than the topology induced by $p.$
2. If $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ is a sequence in $X,$ then $q\left(x_{\bullet }\right):={\left(q\left(x_i\right)\right)}_{i=1}^{\mathrm{\infty }}\to 0$ in $ℝ$ implies $p\left(x_{\bullet }\right)\to 0$ in $ℝ.$
3. If $x_{\bullet }={\left(x_i\right)}_{i\in I}$ is a net in $X,$ then $q\left(x_{\bullet }\right):={\left(q\left(x_i\right)\right)}_{i\in I}\to 0$ in $ℝ$ implies $p\left(x_{\bullet }\right)\to 0$ in $ℝ.$
4. $p$ is bounded on $\{x\in X:q(x)<1\}.$
5. If $inf\{q(x):p(x)=1,x\in X\}=0$ then $p(x)=0$ for all $x\in X.$
6. There exists a real $K>0$ such that $p\le Kq$ on $X.$

The seminorms $p$ and $q$ are called equivalent 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 $x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}$ is a sequence in $X$ then $q\left(x_{\bullet }\right):={\left(q\left(x_i\right)\right)}_{i=1}^{\mathrm{\infty }}\to 0$ if and only if $p\left(x_{\bullet }\right)\to 0.$
4. There exist positive real numbers $r>0$ and $R>0$ such that $rq\le p\le Rq.$

Normability and seminormability

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. 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 $X_b^{\prime }$ of $X$ is normable.
5. The strong dual $X_b^{\prime }$ of $X$ is metrizable.

Furthermore, $X$ is finite dimensional if and only if $X_{\sigma }^{\prime }$ is normable (here $X_{\sigma }^{\prime }$ denotes $X^{\prime }$ 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 $\{x\in X:p(x)<r\}$ in $X$ is equal to $\{x\in X:p(x)\le r\}.$
• The closure of $\{0\}$ in a locally convex space $X$ whose topology is defined by a family of continuous seminorms $𝒫$ is equal to $\bigcap _{p\in 𝒫}{p}^{-1}(0).$
• 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. $\{x\in X:p(x)<1\}$ is open in $X$;
4. $\{x\in X:p(x)\le 1\}$ 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 $p\le q.$

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 $f\le p$ 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 $$\Vert F{\Vert }_{p,q}:=sup\{q(F(x)):p(x)\le 1,x\in X\}.$$

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

1. $F$ is continuous;
2. $\Vert F{\Vert }_{p,q}<\mathrm{\infty }$;
3. There exists a real $K\ge 0$ such that $p\le Kq$;
   • In this case, $\Vert F{\Vert }_{p,q}\le K.$

If $F$ is continuous then $q(F(x))\le \Vert F{\Vert }_{p,q}p(x)$ 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 $\Vert F{\Vert }_{p,q}.$ This seminorm is a norm if $q$ is a norm.

Generalizations

The concept of norm in composition algebras does not share the usual properties of a norm.

A composition algebra $(A,\ast ,N)$ consists of an algebra over a field $A,$ an involution $\ast ,$ 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 ultraseminorm or a non-Archimedean seminorm is a seminorm $p:X\to ℝ$ that also satisfies $p(x+y)\le max\{p(x),p(y)\}\text{ for all }x,y\in X.$

Weakening subadditivity: Quasi-seminorms

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

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

Weakening homogeneity - $k$-seminorms

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

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

Suppose that $q$ is a quasi-seminorm on a vector space $X$ with multiplier $b.$ If $0<\sqrt{k}<{\log }_{2}b$ then there exists $k$-seminorm $p$ on $X$ equivalent to $q.$

See also

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Notes

Proofs

References

• Template:Adasch Topological Vector Spaces
• Template:Berberian Lectures in Functional Analysis and Operator Theory
• Template:Bourbaki Topological Vector Spaces
• Template:Conway A Course in Functional Analysis
• Template:Edwards Functional Analysis Theory and Applications
• Template:Grothendieck Topological Vector Spaces
• Template:Jarchow Locally Convex Spaces
• Template:Khaleelulla Counterexamples in Topological Vector Spaces
• Template:Köthe Topological Vector Spaces I
• Template:Kubrusly The Elements of Operator Theory 2nd Edition 2011
• Template:Narici Beckenstein Topological Vector Spaces
• Prugovečki, Eduard (1981). Quantum mechanics in Hilbert space (2nd ed.). Academic Press. p. 20. ISBN 0-12-566060-X.
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Schechter Handbook of Analysis and Its Foundations
• Template:Swartz An Introduction to Functional Analysis
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Wilansky Modern Methods in Topological Vector Spaces

External links

• Sublinear functions
• The sandwich theorem for sublinear and super linear functionals