Seminorm

{{Short description|Mathematical function}} 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 <math>X</math> be a vector space over either the real numbers <math>\R</math> or the complex numbers <math>\Complex.</math> A real-valued function <math>p : X \to \R</math> is called a {{em|seminorm}} if it satisfies the following two conditions:

# Subadditivity{{sfn|Kubrusly|2011|p=200}}/Triangle inequality: <math>p(x + y) \leq p(x) + p(y)</math> for all <math>x, y \in X.</math> # Absolute homogeneity:{{sfn|Kubrusly|2011|p=200}} <math>p(s x) =|s|p(x)</math> for all <math>x \in X</math> and all scalars <math>s.</math>

These two conditions imply that <math>p(0) = 0</math><ref group="proof">If <math>z \in X</math> denotes the zero vector in <math>X</math> while <math>0</math> denote the zero scalar, then absolute homogeneity implies that <math>p(z) = p(0 z) = |0|p(z) = 0 p(z) = 0.</math> <math>\blacksquare</math></ref> and that every seminorm <math>p</math> also has the following property:<ref group="proof">Suppose <math>p : X \to \R</math> is a seminorm and let <math>x \in X.</math> Then absolute homogeneity implies <math>p(-x) = p((-1) x) =|-1|p(x) = p(x).</math> The triangle inequality now implies <math>p(0) = p(x + (- x)) \leq p(x) + p(-x) = p(x) + p(x) = 2 p(x).</math> Because <math>x</math> was an arbitrary vector in <math>X,</math> it follows that <math>p(0) \leq 2 p(0),</math> which implies that <math>0 \leq p(0)</math> (by subtracting <math>p(0)</math> from both sides). Thus <math>0 \leq p(0) \leq 2 p(x)</math> which implies <math>0 \leq p(x)</math> (by multiplying through by <math>1/2</math>). <math>\blacksquare</math></ref> <ol start=3> <li>Nonnegativity:{{sfn|Kubrusly|2011|p=200}} <math>p(x) \geq 0</math> for all <math>x \in X.</math></li> </ol>

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 <math>X</math> is a seminorm that also separates points, meaning that it has the following additional property: <ol start=4> <li>Positive definite/Positive{{sfn|Kubrusly|2011|p=200}}/{{visible anchor|Point-separating}}: whenever <math>x \in X</math> satisfies <math>p(x) = 0,</math> then <math>x = 0.</math></li> </ol>

A {{em|{{visible anchor|seminormed space}}}} is a pair <math>(X, p)</math> consisting of a vector space <math>X</math> and a seminorm <math>p</math> on <math>X.</math> If the seminorm <math>p</math> is also a norm then the seminormed space <math>(X, p)</math> 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 <math>p : X \to \R</math> 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 <math>p : X \to \R</math> is a seminorm if and only if it is a sublinear and balanced function.

==Examples==

<ul> <li>The {{em|trivial seminorm}} on <math>X,</math> which refers to the constant <math>0</math> map on <math>X,</math> induces the indiscrete topology on <math>X.</math></li> <li>Let <math>\mu</math> be a measure on a space <math>\Omega</math>. For an arbitrary constant <math>c \geq 1</math>, let <math>X</math> be the set of all functions <math>f: \Omega \rightarrow \mathbb{R}</math> for which <math display="block">\lVert f \rVert_c := \left( \int_{\Omega}| f |^c \, d\mu \right)^{1/c}</math> exists and is finite. It can be shown that <math>X</math> is a vector space, and the functional <math>\lVert \cdot \rVert_c</math> is a seminorm on <math>X</math>. However, it is not always a norm (e.g. if <math>\Omega = \mathbb{R}</math> and <math>\mu</math> is the Lebesgue measure) because <math>\lVert h \rVert_c = 0</math> does not always imply <math>h = 0</math>. To make <math>\lVert \cdot \rVert_c</math> a norm, quotient <math>X</math> by the closed subspace of functions <math>h</math> with <math>\lVert h \rVert_c = 0</math>. The resulting space, <math>L^c(\mu)</math>, has a norm induced by <math>\lVert \cdot \rVert_c</math>.</li> <li>If <math>f</math> is any linear form on a vector space then its absolute value <math>|f|,</math> defined by <math>x \mapsto |f(x)|,</math> is a seminorm.</li> <li>A sublinear function <math>f : X \to \R</math> on a real vector space <math>X</math> is a seminorm if and only if it is a {{em|symmetric function}}, meaning that <math>f(-x) = f(x)</math> for all <math>x \in X.</math></li> <li>Every real-valued sublinear function <math>f : X \to \R</math> on a real vector space <math>X</math> induces a seminorm <math>p : X \to \R</math> defined by <math>p(x) := \max \{f(x), f(-x)\}.</math>{{sfn|Narici|Beckenstein|2011|pp=120–121}}</li> <li>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).</li> <li>If <math>p : X \to \R</math> and <math>q : Y \to \R</math> are seminorms (respectively, norms) on <math>X</math> and <math>Y</math> then the map <math>r : X \times Y \to \R</math> defined by <math>r(x, y) = p(x) + q(y)</math> is a seminorm (respectively, a norm) on <math>X \times Y.</math> In particular, the maps on <math>X \times Y</math> defined by <math>(x, y) \mapsto p(x)</math> and <math>(x, y) \mapsto q(y)</math> are both seminorms on <math>X \times Y.</math></li> <li>If <math>p</math> and <math>q</math> are seminorms on <math>X</math> then so are{{sfn|Narici|Beckenstein|2011|pp=116–128}} <math display="block">(p \vee q)(x) = \max \{p(x), q(x)\}</math> and <math display="block">(p \wedge q)(x) := \inf \{p(y) + q(z) : x = y + z \text{ with } y, z \in X\}</math> where <math>p \wedge q \leq p</math> and <math>p \wedge q \leq q.</math>{{sfn|Wilansky|2013|pp=15-21}} </li> <li>The space of seminorms on <math>X</math> is generally not a distributive lattice with respect to the above operations. For example, over <math>\R^2</math>, <math>p(x, y) := \max(|x|, |y|), q(x, y) := 2|x|, r(x, y) := 2|y| </math> are such that <math display="block">((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\}</math> while <math>(p \vee q \wedge r) (x, y) := \max(|x|, |y|)</math></li> <li>If <math>L : X \to Y</math> is a linear map and <math>q : Y \to \R</math> is a seminorm on <math>Y,</math> then <math>q \circ L : X \to \R</math> is a seminorm on <math>X.</math> The seminorm <math>q \circ L</math> will be a norm on <math>X</math> if and only if <math>L</math> is injective and the restriction <math>q\big\vert_{L(X)}</math> is a norm on <math>L(X).</math></li> </ul>

==Minkowski functionals and seminorms== {{Main|Minkowski functional}}

Seminorms on a vector space <math>X</math> are intimately tied, via Minkowski functionals, to subsets of <math>X</math> that are convex, balanced, and absorbing. Given such a subset <math>D</math> of <math>X,</math> the Minkowski functional of <math>D</math> is a seminorm. Conversely, given a seminorm <math>p</math> on <math>X,</math> the sets<math>\{x \in X : p(x) < 1\}</math> and <math>\{x \in X : p(x) \leq 1\}</math> 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 <math>p.</math>{{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, <math>p(0) = 0,</math> and for all vectors <math>x, y \in X</math>: the reverse triangle inequality: {{sfn|Narici|Beckenstein|2011|pp=120-121}}{{sfn|Narici|Beckenstein|2011|pp=177-220}} <math display=block>|p(x) - p(y)| \leq p(x - y)</math> and also <math display=inline>0 \leq \max \{p(x), p(-x)\}</math> and <math>p(x) - p(y) \leq p(x - y).</math>{{sfn|Narici|Beckenstein|2011|pp=120-121}}{{sfn|Narici|Beckenstein|2011|pp=177-220}}

For any vector <math>x \in X</math> and positive real <math>r > 0:</math>{{sfn|Narici|Beckenstein|2011|pp=116−128}} <math display=block>x + \{y \in X : p(y) < r\} = \{y \in X : p(x - y) < r\}</math> and furthermore, <math>\{x \in X : p(x) < r\}</math> is an absorbing disk in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}

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

'''Other properties of seminorms'''

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

If <math>p : X \to [0, \infty)</math> is a seminorm on <math>X</math> then <math>\ker p := p^{-1}(0)</math> is a vector subspace of <math>X</math> and for every <math>x \in X,</math> <math>p</math> is constant on the set <math>x + \ker p = \{x + k : p(k) = 0\}</math> and equal to <math>p(x).</math><ref group=proof name=ConstantOnEquivClasses>Let <math>x \in X</math> and <math>k \in p^{-1}(0).</math> It remains to show that <math>p(x + k) = p(x).</math> The triangle inequality implies <math>p(x + k) \leq p(x) + p(k) = p(x) + 0 = p(x).</math> Since <math>p(-k) = 0,</math> <math>p(x) = p(x) - p(-k) \leq p(x - (-k)) = p(x + k),</math> as desired. <math>\blacksquare</math></ref>

Furthermore, for any real <math>r > 0,</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}} <math display="block">r \{x \in X : p(x) < 1\} = \{x \in X : p(x) < r\} = \left\{x \in X : \tfrac{1}{r} p(x) < 1 \right\}.</math>

If <math>D</math> is a set satisfying <math>\{x \in X : p(x) < 1\} \subseteq D \subseteq \{x \in X : p(x) \leq 1\}</math> then <math>D</math> is absorbing in <math>X</math> and <math>p = p_D</math> where <math>p_D</math> denotes the Minkowski functional associated with <math>D</math> (that is, the gauge of <math>D</math>).{{sfn|Schaefer|Wolff|1999|p=40}} In particular, if <math>D</math> is as above and <math>q</math> is any seminorm on <math>X,</math> then <math>q = p</math> if and only if <math>\{x \in X : q(x) < 1\} \subseteq D \subseteq \{x \in X : q(x) \leq\}.</math>{{sfn|Schaefer|Wolff|1999|p=40}}

If <math>(X, \|\,\cdot\,\|)</math> is a normed space and <math>x, y \in X</math> then <math>\|x - y\| = \|x - z\| + \|z - y\|</math> for all <math>z</math> in the interval <math>[x, y].</math>{{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 <math>p : X \to \R</math> be a non-negative function. The following are equivalent: <ol> <li><math>p</math> is a seminorm.</li> <li><math>p</math> is a convex <math>F</math>-seminorm.</li> <li><math>p</math> is a convex balanced ''G''-seminorm.{{sfn|Schechter|1996|p=691}}</li> </ol>

If any of the above conditions hold, then the following are equivalent: <ol> <li><math>p</math> is a norm;</li> <li><math>\{x \in X : p(x) < 1\}</math> does not contain a non-trivial vector subspace.{{sfn|Narici|Beckenstein|2011|p=149}}</li> <li>There exists a norm on <math>X,</math> with respect to which, <math>\{x \in X : p(x) < 1\}</math> is bounded.</li> </ol>

If <math>p</math> is a sublinear function on a real vector space <math>X</math> then the following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=177-220}} <ol> <li><math>p</math> is a linear functional;</li> <li><math>p(x) + p(-x) \leq 0 \text{ for every } x \in X</math>;</li> <li><math>p(x) + p(-x) = 0 \text{ for every } x \in X</math>;</li> </ol>

===Inequalities involving seminorms===

If <math>p, q : X \to [0, \infty)</math> are seminorms on <math>X</math> then: <ul> <li><math>p \leq q</math> if and only if <math>q(x) \leq 1</math> implies <math>p(x) \leq 1.</math>{{sfn|Narici|Beckenstein|2011|pp=149–153}}</li> <li>If <math>a > 0</math> and <math>b > 0</math> are such that <math>p(x) < a</math> implies <math>q(x) \leq b,</math> then <math>a q(x) \leq b p(x)</math> for all <math>x \in X.</math> {{sfn|Wilansky|2013|pp=18-21}}</li> <li>Suppose <math>a</math> and <math>b</math> are positive real numbers and <math>q, p_1, \ldots, p_n</math> are seminorms on <math>X</math> such that for every <math>x \in X,</math> if <math>\max \{p_1(x), \ldots, p_n(x)\} < a</math> then <math>q(x) < b.</math> Then <math>a q \leq b \left(p_1 + \cdots + p_n\right).</math>{{sfn|Narici|Beckenstein|2011|p=149}}</li> <li>If <math>X</math> is a vector space over the reals and <math>f</math> is a non-zero linear functional on <math>X,</math> then <math>f \leq p</math> if and only if <math>\varnothing = f^{-1}(1) \cap \{x \in X : p(x) < 1\}.</math>{{sfn|Narici|Beckenstein|2011|pp=149–153}}</li> </ul>

If <math>p</math> is a seminorm on <math>X</math> and <math>f</math> is a linear functional on <math>X</math> then: <ul> <li><math>|f| \leq p</math> on <math>X</math> if and only if <math>\operatorname{Re} f \leq p</math> on <math>X</math> (see footnote for proof).<ref>Obvious if <math>X</math> is a real vector space. For the non-trivial direction, assume that <math>\operatorname{Re} f \leq p</math> on <math>X</math> and let <math>x \in X.</math> Let <math>r \geq 0</math> and <math>t</math> be real numbers such that <math>f(x) = r e^{i t}.</math> Then <math>|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).</math></ref>{{sfn|Wilansky|2013|p=20}}</li> <li><math>f \leq p</math> on <math>X</math> if and only if <math>f^{-1}(1) \cap \{x \in X : p(x) < 1 = \varnothing\}.</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}{{sfn|Narici|Beckenstein|2011|pp=149–153}}</li> <li>If <math>a > 0</math> and <math>b > 0</math> are such that <math>p(x) < a</math> implies <math>f(x) \neq b,</math> then <math>a |f(x)| \leq b p(x)</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013|pp=18-21}}</li> </ul>

===Hahn–Banach theorem for seminorms===

Seminorms offer a particularly clean formulation of the Hahn–Banach theorem: :If <math>M</math> is a vector subspace of a seminormed space <math>(X, p)</math> and if <math>f</math> is a continuous linear functional on <math>M,</math> then <math>f</math> may be extended to a continuous linear functional <math>F</math> on <math>X</math> that has the same norm as <math>f.</math>{{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 <math>M</math> is a vector subspace of <math>X,</math> <math>p</math> is a seminorm on <math>M,</math> and <math>q</math> is a seminorm on <math>X</math> such that <math>p \leq q\big\vert_M,</math> then there exists a seminorm <math>P</math> on <math>X</math> such that <math>P\big\vert_M = p</math> and <math>P \leq q.</math> }}

:'''Proof''': Let <math>S</math> be the convex hull of <math>\{m \in M : p(m) \leq 1\} \cup \{x \in X : q(x) \leq 1\}.</math> Then <math>S</math> is an absorbing disk in <math>X</math> and so the Minkowski functional <math>P</math> of <math>S</math> is a seminorm on <math>X.</math> This seminorm satisfies <math>p = P</math> on <math>M</math> and <math>P \leq q</math> on <math>X.</math> <math>\blacksquare</math>

==Topologies of seminormed spaces==

===Pseudometrics and the induced topology===

A seminorm <math>p</math> on <math>X</math> induces a topology, called the {{em|seminorm-induced topology}}, via the canonical translation-invariant pseudometric <math>d_p : X \times X \to \R</math>; <math>d_p(x, y) := p(x - y) = p(y - x).</math> This topology is Hausdorff if and only if <math>d_p</math> is a metric, which occurs if and only if <math>p</math> is a norm.{{sfn|Wilansky|2013 |pp=15-21}} This topology makes <math>X</math> 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: <math display=block>\{x \in X : p(x) < r\} \quad \text{ or } \quad \{x \in X : p(x) \leq r\}</math> as <math>r > 0</math> ranges over the positive reals. Every seminormed space <math>(X, p)</math> 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 <math>X</math> with seminorm <math>p</math> induces a vector space quotient <math>X / W,</math> where <math>W</math> is the subspace of <math>X</math> consisting of all vectors <math>x \in X</math> with <math>p(x) = 0.</math> Then <math>X / W</math> carries a norm defined by <math>p(x + W) = p(x).</math> The resulting topology, pulled back to <math>X,</math> is precisely the topology induced by <math>p.</math>

Any seminorm-induced topology makes <math>X</math> locally convex, as follows. If <math>p</math> is a seminorm on <math>X</math> and <math>r \in \R,</math> call the set <math>\{x \in X : p(x) < r\}</math> the {{em|open ball of radius <math>r</math> about the origin}}; likewise the closed ball of radius <math>r</math> is <math>\{x \in X : p(x) \leq r\}.</math> The set of all open (resp. closed) <math>p</math>-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the <math>p</math>-topology on <math>X.</math>

====Stronger, weaker, and equivalent seminorms====

The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If <math>p</math> and <math>q</math> are seminorms on <math>X,</math> then we say that <math>q</math> is {{em|stronger}} than <math>p</math> and that <math>p</math> is {{em|weaker}} than <math>q</math> if any of the following equivalent conditions holds:

# The topology on <math>X</math> induced by <math>q</math> is finer than the topology induced by <math>p.</math> # If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a sequence in <math>X,</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0</math> in <math>\R</math> implies <math>p\left(x_{\bull}\right) \to 0</math> in <math>\R.</math>{{sfn|Wilansky|2013 |pp=15-21}} # If <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is a net in <math>X,</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i \in I} \to 0</math> in <math>\R</math> implies <math>p\left(x_{\bull}\right) \to 0</math> in <math>\R.</math> # <math>p</math> is bounded on <math>\{x \in X : q(x) < 1\}.</math>{{sfn|Wilansky|2013 |pp=15-21}} # If <math>\inf{} \{q(x) : p(x) = 1, x \in X\} = 0</math> then <math>p(x) = 0</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013 |pp=15-21}} # There exists a real <math>K > 0</math> such that <math>p \leq K q</math> on <math>X.</math>{{sfn|Wilansky|2013 |pp=15-21}}

The seminorms <math>p</math> and <math>q</math> 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: <ol> <li>The topology on <math>X</math> induced by <math>q</math> is the same as the topology induced by <math>p.</math></li> <li><math>q</math> is stronger than <math>p</math> and <math>p</math> is stronger than <math>q.</math>{{sfn|Wilansky|2013|pp=15-21}}</li> <li>If <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> is a sequence in <math>X</math> then <math>q\left(x_{\bull}\right) := \left(q\left(x_i\right)\right)_{i=1}^{\infty} \to 0</math> if and only if <math>p\left(x_{\bull}\right) \to 0.</math></li> <li>There exist positive real numbers <math>r > 0</math> and <math>R > 0</math> such that <math>r q \leq p \leq R q.</math></li> </ol>

===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 <math>X</math> is a Hausdorff locally convex TVS then the following are equivalent: <ol> <li><math>X</math> is normable.</li> <li><math>X</math> is seminormable.</li> <li><math>X</math> has a bounded neighborhood of the origin.</li> <li>The strong dual <math>X^{\prime}_b</math> of <math>X</math> is normable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}</li> <li>The strong dual <math>X^{\prime}_b</math> of <math>X</math> is metrizable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}</li> </ol> Furthermore, <math>X</math> is finite dimensional if and only if <math>X^{\prime}_{\sigma}</math> is normable (here <math>X^{\prime}_{\sigma}</math> denotes <math>X^{\prime}</math> 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===

<ul> <li>If <math>X</math> is a TVS and <math>p</math> is a continuous seminorm on <math>X,</math> then the closure of <math>\{x \in X : p(x) < r\}</math> in <math>X</math> is equal to <math>\{x \in X : p(x) \leq r\}.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li> <li>The closure of <math>\{0\}</math> in a locally convex space <math>X</math> whose topology is defined by a family of continuous seminorms <math>\mathcal{P}</math> is equal to <math>\bigcap_{p \in \mathcal{P}} p^{-1}(0).</math>{{sfn|Narici|Beckenstein|2011|pp=149-153}}</li> <li>A subset <math>S</math> in a seminormed space <math>(X, p)</math> is bounded if and only if <math>p(S)</math> is bounded.{{sfn|Wilansky|2013|pp=49-50}}</li> <li>If <math>(X, p)</math> is a seminormed space then the locally convex topology that <math>p</math> induces on <math>X</math> makes <math>X</math> into a pseudometrizable TVS with a canonical pseudometric given by <math>d(x, y) := p(x - y)</math> for all <math>x, y \in X.</math>{{sfn|Narici|Beckenstein|2011|pp=115-154}}</li> <li>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}}</li> </ul>

===Continuity of seminorms===

If <math>p</math> is a seminorm on a topological vector space <math>X,</math> then the following are equivalent:{{sfn|Schaefer|Wolff|1999|p=40}} <ol> <li><math>p</math> is continuous.</li> <li><math>p</math> is continuous at 0;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li> <li><math>\{x \in X : p(x) < 1\}</math> is open in <math>X</math>;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li> <li><math>\{x \in X : p(x) \leq 1\}</math> is closed neighborhood of 0 in <math>X</math>;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li> <li><math>p</math> is uniformly continuous on <math>X</math>;{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li> <li>There exists a continuous seminorm <math>q</math> on <math>X</math> such that <math>p \leq q.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}</li> </ol>

In particular, if <math>(X, p)</math> is a seminormed space then a seminorm <math>q</math> on <math>X</math> is continuous if and only if <math>q</math> is dominated by a positive scalar multiple of <math>p.</math>{{sfn|Narici|Beckenstein|2011|pp=116–128}}

If <math>X</math> is a real TVS, <math>f</math> is a linear functional on <math>X,</math> and <math>p</math> is a continuous seminorm (or more generally, a sublinear function) on <math>X,</math> then <math>f \leq p</math> on <math>X</math> implies that <math>f</math> is continuous.{{sfn|Narici|Beckenstein|2011|pp=177-220}}

===Continuity of linear maps===

If <math>F : (X, p) \to (Y, q)</math> is a map between seminormed spaces then let{{sfn|Wilansky|2013|pp=21-26}} <math display="block">\|F\|_{p,q} := \sup \{q(F(x)) : p(x) \leq 1, x \in X\}.</math>

If <math>F : (X, p) \to (Y, q)</math> is a linear map between seminormed spaces then the following are equivalent: <ol> <li><math>F</math> is continuous;</li> <li><math>\|F\|_{p,q} < \infty</math>;{{sfn|Wilansky|2013|pp=21-26}}</li> <li>There exists a real <math>K \geq 0</math> such that <math>p \leq K q</math>;{{sfn|Wilansky|2013|pp=21-26}} * In this case, <math>\|F\|_{p,q} \leq K.</math></li> </ol> If <math>F</math> is continuous then <math>q(F(x)) \leq \|F\|_{p,q} p(x)</math> for all <math>x \in X.</math>{{sfn|Wilansky|2013|pp=21-26}}

The space of all continuous linear maps <math>F : (X, p) \to (Y, q)</math> between seminormed spaces is itself a seminormed space under the seminorm <math>\|F\|_{p,q}.</math> This seminorm is a norm if <math>q</math> 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 <math>(A, *, N)</math> consists of an algebra over a field <math>A,</math> an involution <math>\,*,</math> and a quadratic form <math>N,</math> which is called the "norm". In several cases <math>N</math> is an isotropic quadratic form so that <math>A</math> 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 <math>p : X \to \R</math> that also satisfies <math>p(x + y) \leq \max \{p(x), p(y)\} \text{ for all } x, y \in X.</math>

'''Weakening subadditivity: Quasi-seminorms'''

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

A quasi-seminorm that separates points is called a {{em|quasi-norm}} on <math>X.</math>

'''Weakening homogeneity - <math>k</math>-seminorms'''

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

We have the following relationship between quasi-seminorms and <math>k</math>-seminorms: {{block indent | em = 1.5 | text = Suppose that <math>q</math> is a quasi-seminorm on a vector space <math>X</math> with multiplier <math>b.</math> If <math>0 < \sqrt{k} < \log_2 b</math> then there exists <math>k</math>-seminorm <math>p</math> on <math>X</math> equivalent to <math>q.</math>}}

==See also==

* {{annotated link|Asymmetric norm}} * {{annotated link|Banach space}} * {{annotated link|Contraction mapping}} * {{annotated link|Finest locally convex topology}} * {{annotated link|Hahn-Banach theorem}} * {{annotated link|Gowers norm}} * {{annotated link|Locally convex topological vector space}} * {{annotated link|Mahalanobis distance}} * {{annotated link|Matrix norm}} * {{annotated link|Minkowski functional}} * {{annotated link|Norm (mathematics)}} * {{annotated link|Normed vector space}} * {{annotated link|Relation of norms and metrics}} * {{annotated link|Sublinear function}}

==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]

{{DEFAULTSORT:Norm (Mathematics)}} Category:Norms (mathematics) Category:Linear algebra