seminorm

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 $\Complex.$

A real-valued function $p : X \to \R$ is called a {{em|seminorm}} if it satisfies the following two conditions:

Subadditivity{{sfn|Kubrusly|2011|p=200}}/Triangle inequality: $p(x + y) \leq p(x) + p(y)$ for all $x, y \in X.$
Absolute homogeneity:{{sfn|Kubrusly|2011|p=200}} $p(s x) =|s|p(x)$ 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 $1/2$ ). $\blacksquare$

Nonnegativity:{{sfn|Kubrusly|2011|p=200}} $p(x) \geq 0$ 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:

Positive definite/Positive{{sfn|Kubrusly|2011|p=200}}/{{visible anchor|Point-separating}}: whenever $x \in X$ satisfies $p(x) = 0,$ then $x = 0.$

A {{em|{{visible anchor|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 {{em|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 {{em|sublinear function}} if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is {{em|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 {{em|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 $c \geq 1$ , let $X$ be the set of all functions $f: \Omega \rightarrow \mathbb{R}$ for which

$\lVert f \rVert_c := \left( \int_{\Omega}| f |^c \, d\mu \right)^{1/c}$

exists and is finite. It can be shown that $X$ is a vector space, and the functional $\lVert \cdot \rVert_c$ is a seminorm on $X$ . However, it is not always a norm (e.g. if $\Omega = \mathbb{R}$ and $\mu$ is the Lebesgue measure) because $\lVert h \rVert_c = 0$ does not always imply $h = 0$ . To make $\lVert \cdot \rVert_c$ a norm, quotient $X$ by the closed subspace of functions $h$ with $\lVert h \rVert_c = 0$ . The resulting space, $L^c(\mu)$ , has a norm induced by $\lVert \cdot \rVert_c$ .

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 {{em|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 $p(x) := \max \{f(x), f(-x)\}.$ {{sfn|Narici|Beckenstein|2011|pp=120–121}}

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{{sfn|Narici|Beckenstein|2011|pp=116–128}}

$(p \vee q)(x) = \max \{p(x), q(x)\}$ and $(p \wedge q)(x) := \inf \{p(y) + q(z) : x = y + z \text{ with } y, z \in X\}$

where $p \wedge q \leq p$ and $p \wedge q \leq q.$ {{sfn|Wilansky|2013|pp=15-21}}

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 $q\big\vert_{L(X)}$ 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 $\{x \in X : p(x) \leq 1\}$ 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 $p.$ {{sfn|Schaefer|Wolff|1999|p=40}}

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: {{sfn|Narici|Beckenstein|2011|pp=120-121}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}

$|p(x) - p(y)| \leq p(x - y)$

and also

$0 \leq \max \{p(x), p(-x)\}$ and $p(x) - p(y) \leq p(x - y).$ {{sfn|Narici|Beckenstein|2011|pp=120-121}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}

For any vector $x \in X$ and positive real $r > 0:$ {{sfn|Narici|Beckenstein|2011|pp=116−128}}

$x + \{y \in X : p(y) < r\} = \{y \in X : p(x - y) < r\}$

and furthermore, $\{x \in X : p(x) < r\}$ is an absorbing disk in $X.$ {{sfn|Narici|Beckenstein|2011|pp=116–128}}

If $p$ is a sublinear function on a real vector space $X$ then there exists a linear functional $f$ on $X$ such that $f \leq p$ {{sfn|Narici|Beckenstein|2011|pp=177-220}} and furthermore, for any linear functional $g$ on $X,$ $g \leq p$ on $X$ if and only if $g^{-1}(1) \cap \{x \in X : p(x) < 1\} = \varnothing.$ {{sfn|Narici|Beckenstein|2011|pp=177-220}}

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 $p : X \to [0, \infty)$ 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 $p(x).$ Let $x \in X$ and $k \in p^{-1}(0).$ It remains to show that $p(x + k) = p(x).$ The triangle inequality implies $p(x + k) \leq p(x) + p(k) = p(x) + 0 = p(x).$ Since $p(-k) = 0,$ $p(x) = p(x) - p(-k) \leq p(x - (-k)) = p(x + k),$ as desired. $\blacksquare$

Furthermore, for any real $r > 0,$ {{sfn|Narici|Beckenstein|2011|pp=116–128}}

$r \{x \in X : p(x) < 1\} = \{x \in X : p(x) < r\} = \left\{x \in X : \tfrac{1}{r} p(x) < 1 \right\}.$

If $D$ is a set satisfying $\{x \in X : p(x) < 1\} \subseteq D \subseteq \{x \in X : p(x) \leq 1\}$ 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 $D$ ).{{sfn|Schaefer|Wolff|1999|p=40}} In particular, if $D$ is as above and $q$ is any seminorm on $X,$ then $q = p$ if and only if $\{x \in X : q(x) < 1\} \subseteq D \subseteq \{x \in X : q(x) \leq\}.$ {{sfn|Schaefer|Wolff|1999|p=40}}

If $(X, \|\,\cdot\,\|)$ is a normed space and $x, y \in X$ then $\|x - y\| = \|x - z\| + \|z - y\|$ for all $z$ in the interval $[x, y].$ {{sfn|Narici|Beckenstein|2011|pp=107-113}}

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:

$p$ is a seminorm.

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

$p$ is a convex balanced G-seminorm.{{sfn|Schechter|1996|p=691}}

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

$p$ is a norm;

$\{x \in X : p(x) < 1\}$ does not contain a non-trivial vector subspace.{{sfn|Narici|Beckenstein|2011|p=149}}

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:{{sfn|Narici|Beckenstein|2011|pp=177-220}}

$p$ is a linear functional;

$p(x) + p(-x) \leq 0 \text{ for every } x \in X$ ;

$p(x) + p(-x) = 0 \text{ for every } x \in X$ ;

=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 $p(x) \leq 1.$ {{sfn|Narici|Beckenstein|2011|pp=149–153}}

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 $x \in X.$ {{sfn|Wilansky|2013|pp=18-21}}

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 $\max \{p_1(x), \ldots, p_n(x)\} < a$ then $q(x) < b.$ Then $a q \leq b \left(p_1 + \cdots + p_n\right).$ {{sfn|Narici|Beckenstein|2011|p=149}}

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 $\varnothing = f^{-1}(1) \cap \{x \in X : p(x) < 1\}.$ {{sfn|Narici|Beckenstein|2011|pp=149–153}}

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 $\operatorname{Re} f \leq p$ on $X$ (see footnote for proof).Obvious if $X$ is a real vector space. For the non-trivial direction, assume that $\operatorname{Re} f \leq p$ on $X$ and let $x \in X.$ Let $r \geq 0$ and $t$ be real numbers such that $f(x) = r e^{i t}.$ Then $|f(x)|= r = f\left(e^{-it} x\right) = \operatorname{Re}\left(f\left(e^{-it} x\right)\right) \leq p\left(e^{-it} x\right) = p(x).$ {{sfn|Wilansky|2013|p=20}}

$f \leq p$ on $X$ if and only if $f^{-1}(1) \cap \{x \in X : p(x) < 1 = \varnothing\}.$ {{sfn|Narici|Beckenstein|2011|pp=177-220}}{{sfn|Narici|Beckenstein|2011|pp=149–153}}

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 $x \in X.$ {{sfn|Wilansky|2013|pp=18-21}}

=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 $f.$ {{sfn|Wilansky|2013|pp=21-26}}

A similar extension property also holds for seminorms:

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=150}}{{sfn|Wilansky|2013|pp=18-21}}|note=Extending seminorms|math_statement=

If $M$ is a vector subspace of $X,$ $p$ is a seminorm on $M,$ and $q$ is a seminorm on $X$ such that $p \leq q\big\vert_M,$ then there exists a seminorm $P$ on $X$ such that $P\big\vert_M = p$ and $P \leq q.$

}}

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

Topologies of seminormed spaces

=Pseudometrics and the induced topology=

A seminorm $p$ on $X$ induces a topology, called the {{em|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.{{sfn|Wilansky|2013 |pp=15-21}}

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:

$\{x \in X : p(x) < r\} \quad \text{ or } \quad \{x \in X : p(x) \leq r\}$

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 {{em|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 {{em|open ball of radius $r$ about the origin}}; likewise the closed ball of radius $r$ is $\{x \in X : p(x) \leq r\}.$ 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 {{em|stronger}} than $p$ and that $p$ is {{em|weaker}} 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 $x_{\bull} = \left(x_i\right)_{i=1}^{\infty}$ is a sequence in $X,$ then $q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0$ in $\R$ implies $p\left(x_{\bull}\right) \to 0$ in $\R.$ {{sfn|Wilansky|2013 |pp=15-21}}
If $x_{\bull} = \left(x_i\right)_{i \in I}$ is a net in $X,$ then $q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i \in I} \to 0$ in $\R$ implies $p\left(x_{\bull}\right) \to 0$ in $\R.$
$p$ is bounded on $\{x \in X : q(x) < 1\}.$ {{sfn|Wilansky|2013 |pp=15-21}}
If $\inf{} \{q(x) : p(x) = 1, x \in X\} = 0$ then $p(x) = 0$ for all $x \in X.$ {{sfn|Wilansky|2013 |pp=15-21}}
There exists a real $K > 0$ such that $p \leq K q$ on $X.$ {{sfn|Wilansky|2013 |pp=15-21}}

The seminorms $p$ and $q$ are called {{em|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 $X$ induced by $q$ is the same as the topology induced by $p.$

$q$ is stronger than $p$ and $p$ is stronger than $q.$ {{sfn|Wilansky|2013|pp=15-21}}

If $x_{\bull} = \left(x_i\right)_{i=1}^{\infty}$ is a sequence in $X$ then $q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0$ if and only if $p\left(x_{\bull}\right) \to 0.$

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

=Normability and seminormability=

{{See also|Normed space|Local boundedness#locally bounded topological vector space}}

A topological vector space (TVS) is said to be a {{em|{{visible anchor|seminormable space}}}} (respectively, a {{em|{{visible anchor|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 {{visible anchor|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.{{sfn|Wilansky|2013|pp=50-51}}

Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

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:

$X$ is normable.

$X$ is seminormable.

$X$ has a bounded neighborhood of the origin.

The strong dual $X^{\prime}_b$ of $X$ is normable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}

The strong dual $X^{\prime}_b$ of $X$ is metrizable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}

Furthermore, $X$ is finite dimensional if and only if $X^{\prime}_{\sigma}$ is normable (here $X^{\prime}_{\sigma}$ 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).{{sfn|Narici|Beckenstein|2011|pp=156–175}}

=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 $\{x \in X : p(x) \leq r\}.$ {{sfn|Narici|Beckenstein|2011|pp=116–128}}

The closure of $\{0\}$ in a locally convex space $X$ whose topology is defined by a family of continuous seminorms $\mathcal{P}$ is equal to $\bigcap_{p \in \mathcal{P}} p^{-1}(0).$ {{sfn|Narici|Beckenstein|2011|pp=149-153}}

A subset $S$ in a seminormed space $(X, p)$ is bounded if and only if $p(S)$ is bounded.{{sfn|Wilansky|2013|pp=49-50}}

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 $x, y \in X.$ {{sfn|Narici|Beckenstein|2011|pp=115-154}}

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).{{sfn|Narici|Beckenstein|2011|pp=156–175}}

=Continuity of seminorms=

If $p$ is a seminorm on a topological vector space $X,$ then the following are equivalent:{{sfn|Schaefer|Wolff|1999|p=40}}

$p$ is continuous.

$p$ is continuous at 0;{{sfn|Narici|Beckenstein|2011|pp=116–128}}

$\{x \in X : p(x) < 1\}$ is open in $X$ ;{{sfn|Narici|Beckenstein|2011|pp=116–128}}

$\{x \in X : p(x) \leq 1\}$ is closed neighborhood of 0 in $X$ ;{{sfn|Narici|Beckenstein|2011|pp=116–128}}

$p$ is uniformly continuous on $X$ ;{{sfn|Narici|Beckenstein|2011|pp=116–128}}

There exists a continuous seminorm $q$ on $X$ such that $p \leq q.$ {{sfn|Narici|Beckenstein|2011|pp=116–128}}

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 $p.$ {{sfn|Narici|Beckenstein|2011|pp=116–128}}

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.{{sfn|Narici|Beckenstein|2011|pp=177-220}}

=Continuity of linear maps=

If $F : (X, p) \to (Y, q)$ is a map between seminormed spaces then let{{sfn|Wilansky|2013|pp=21-26}}

$\|F\|_{p,q} := \sup \{q(F(x)) : p(x) \leq 1, x \in X\}.$

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

$F$ is continuous;

$\|F\|_{p,q} < \infty$ ;{{sfn|Wilansky|2013|pp=21-26}}

There exists a real K \geq 0 such that p \leq K q;{{sfn|Wilansky|2013|pp=21-26}}
- In this case, $\|F\|_{p,q} \leq K.$

If $F$ is continuous then $q(F(x)) \leq \|F\|_{p,q} p(x)$ for all $x \in X.$ {{sfn|Wilansky|2013|pp=21-26}}

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

This seminorm is a norm if $q$ is a norm.{{sfn|Wilansky|2013|pp=21-26}}

Generalizations

The concept of {{em|norm}} in composition algebras does {{em|not}} 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 {{em|ultraseminorm}} or a {{em|non-Archimedean seminorm}} is a seminorm $p : X \to \R$ that also satisfies $p(x + y) \leq \max \{p(x), p(y)\} \text{ for all } x, y \in X.$

Weakening subadditivity: Quasi-seminorms

A map $p : X \to \R$ is called a {{em|quasi-seminorm}} if it is (absolutely) homogeneous and there exists some $b \leq 1$ such that $p(x + y) \leq b p(p(x) + p(y)) \text{ for all } x, y \in X.$

The smallest value of $b$ for which this holds is called the {{em|multiplier of $p.$ }}

A quasi-seminorm that separates points is called a {{em|quasi-norm}} on $X.$

Weakening homogeneity - $k$ -seminorms

A map $p : X \to \R$ is called a {{em| $k$ -seminorm}} 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 {{em| $k$ -norm}} on $X.$

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

{{block indent | em = 1.5 | text = 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 $q.$ }}

Notes

Proofs

References

{{Adasch Topological Vector Spaces}}
{{Berberian Lectures in Functional Analysis and Operator Theory}}
{{Bourbaki Topological Vector Spaces}}
{{Conway A Course in Functional Analysis}}
{{Edwards Functional Analysis Theory and Applications}}
{{Grothendieck Topological Vector Spaces}}
{{Jarchow Locally Convex Spaces}}
{{Khaleelulla Counterexamples in Topological Vector Spaces}}
{{Köthe Topological Vector Spaces I}}
{{Kubrusly The Elements of Operator Theory 2nd Edition 2011}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{cite book|last=Prugovečki|first=Eduard|title=Quantum mechanics in Hilbert space|year=1981|edition=2nd|publisher=Academic Press|page=20|isbn=0-12-566060-X}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Schechter Handbook of Analysis and Its Foundations}}
{{Swartz An Introduction to Functional Analysis}}
{{Trèves François Topological vector spaces, distributions and kernels}}
{{Wilansky Modern Methods in Topological Vector Spaces}}

External links

[https://shodhganga.inflibnet.ac.in/bitstream/10603/13152/9/09_chapter%203.pdf Sublinear functions]
[https://arxiv.org/pdf/1611.02670.pdf The sandwich theorem for sublinear and super linear functionals]

Category:Linear algebra