{"history":[],"article":{"id":"04573e42-ebc8-4cf0-b029-aaf51d83e166","title":"Seminorm","slug":"seminorm","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), particularly in [functional analysis](https://bliptext.com/articles/functional-analysis), a **semi[norm](https://bliptext.com/articles/norm-mathematics)** is a [norm](https://bliptext.com/articles/norm-mathematics) that need not be [positive definite](https://bliptext.com/articles/positive-definite).\nSeminorms are intimately connected with [convex sets](https://bliptext.com/articles/convex-set): every seminorm is the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) of some [absorbing](https://bliptext.com/articles/absorbing-set) [disk](https://bliptext.com/articles/absolutely-convex-set) and, conversely, the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) of any such set is a seminorm.\nA [topological vector space](https://bliptext.com/articles/topological-vector-space) is locally convex if and only if its topology is induced by a family of seminorms.\n\n# Definition\nLet X be a vector space over either the [real numbers](https://bliptext.com/articles/real-number) \\R or the [complex](https://bliptext.com/articles/complex-number) numbers \\Complex.\nA [real-valued function](https://bliptext.com/articles/real-valued-function) is called a if it satisfies the following two conditions:\nThese two conditions imply that p(0) = 0 and that every seminorm p also has the following property:\n

[Nonnegativity](https://bliptext.com/articles/nonnegative): p(x) \\geq 0 for all x \\in X.

\n\nSome 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.\nBy definition, a [norm](https://bliptext.com/articles/norm-mathematics) on X is a semi[norm](https://bliptext.com/articles/norm-mathematics) that also separates points, meaning that it has the following additional property:\n

[Positive definite](https://bliptext.com/articles/positive-definite)/Positive/: whenever x \\in X satisfies p(x) = 0, then x = 0.

\n\nA 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.\nSince absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a [sublinear function](https://bliptext.com/articles/sublinear-function).\nA map is called a if it is subadditive and [positive homogeneous](https://bliptext.com/articles/positive-homogeneous).\nUnlike a seminorm, a sublinear function is necessarily nonnegative.\nSublinear functions are often encountered in the context of the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem).\nA real-valued function is a seminorm if and only if it is a [sublinear](https://bliptext.com/articles/sublinear-function) and [balanced function](https://bliptext.com/articles/balanced-function).\n\n# Examples\n

The on X, which refers to the constant 0 map on X, induces the [indiscrete topology](https://bliptext.com/articles/indiscrete-topology) on X.
Let \\mu be a measure on a space \\Omega.\nFor an arbitrary constant c \\geq 1, let X be the set of all functions for which\nexists and is finite.\nIt can be shown that X is a vector space, and the functional is a seminorm on X. However, it is not always a norm (e.g. if and \\mu is the [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure)) because does not always imply h = 0.\nTo make a norm, quotient X by the closed subspace of functions h with.\nThe [resulting space](https://bliptext.com/articles/lp-space), L^c(\\mu), has a norm induced by .
If f is any [linear form](https://bliptext.com/articles/linear-form) on a vector space then its [absolute value](https://bliptext.com/articles/absolute-value) |f|, defined by is a seminorm.
A [sublinear function](https://bliptext.com/articles/sublinear-function) on a real vector space X is a seminorm if and only if it is a, meaning that for all x \\in X.
Every real-valued [sublinear function](https://bliptext.com/articles/sublinear-function) on a real vector space X induces a seminorm defined by
Any finite sum of seminorms is a seminorm.\nThe restriction of a seminorm (respectively, norm) to a [vector subspace](https://bliptext.com/articles/vector-subspace) is once again a seminorm (respectively, norm).
If and are seminorms (respectively, norms) on X and Y then the map defined by is a seminorm (respectively, a norm) on X \\times Y. In particular, the maps on X \\times Y defined by and are both seminorms on X \\times Y.
If p and q are seminorms on X then so are\nand\nwhere and\n
The space of seminorms on X is generally not a [distributive lattice](https://bliptext.com/articles/distributive-lattice) with respect to the above operations.\nFor example, over \\R^2, are such that\nwhile
If L : X \\to Y is a [linear map](https://bliptext.com/articles/linear-map) and is a seminorm on Y, then 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 is a norm on L(X).

\n\n# Minkowski functionals and seminorms\nSeminorms on a vector space X are intimately tied, via Minkowski functionals, to subsets of X that are [convex](https://bliptext.com/articles/convex-set), [balanced](https://bliptext.com/articles/balanced-set), and [absorbing](https://bliptext.com/articles/absorbing-set).\nGiven such a subset D of X, the Minkowski functional of D is a seminorm.\nConversely, 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.\n\n# Algebraic properties\nEvery seminorm is a [sublinear function](https://bliptext.com/articles/sublinear-function), and thus satisfies all properties of a [sublinear function](https://bliptext.com/articles/sublinear-function), including [convexity](https://bliptext.com/articles/convex-function), p(0) = 0, and for all vectors x, y \\in X: the [reverse triangle inequality](https://bliptext.com/articles/reverse-triangle-inequality):\nand also\nand\nFor any vector x \\in X and positive real r > 0:\nand furthermore, is an [absorbing](https://bliptext.com/articles/absorbing-set) [disk](https://bliptext.com/articles/absolutely-convex-set) in X.\nIf 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 and furthermore, for any linear functional g on X, g \\leq p on X if and only if\n**Other properties of seminorms**\nEvery seminorm is a [balanced function](https://bliptext.com/articles/balanced-function).\nA seminorm p is a norm on X if and only if does not contain a non-trivial vector subspace.\nIf is a seminorm on X then is a vector subspace of X and for every x \\in X, p is constant on the set and equal to p(x).\nFurthermore, for any real r > 0,\nIf D is a set satisfying then D is [absorbing](https://bliptext.com/articles/absorbing-set) in X and p = p_D where p_D denotes the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) associated with D (that is, the gauge of D).\nIn particular, if D is as above and q is any seminorm on X, then q = p if and only if\nIf is a normed space and x, y \\in X then for all z in the interval [x, y].\nEvery norm is a [convex function](https://bliptext.com/articles/convex-function) and consequently, finding a global maximum of a norm-based [objective function](https://bliptext.com/articles/objective-function) is sometimes tractable.\n\n## Relationship to other norm-like concepts\nLet be a non-negative function.\nThe following are equivalent:\n

p is a seminorm.
p is a [convex](https://bliptext.com/articles/convex-function) [F-seminorm](https://bliptext.com/articles/f-seminorm).
p is a convex balanced G-seminorm.

\nIf any of the above conditions hold, then the following are equivalent:\n

p is a norm;
does not contain a non-trivial vector subspace.
There exists a [norm](https://bliptext.com/articles/normed-vector-space) on X, with respect to which, is bounded.

\nIf p is a sublinear function on a real vector space X then the following are equivalent:\n

p is a [linear functional](https://bliptext.com/articles/linear-functional);
;
;

\n\n## Inequalities involving seminorms\nIf are seminorms on X then:\n

p \\leq q if and only if q(x) \\leq 1 implies
If a > 0 and b > 0 are such that p(x) < a implies then for all x \\in X.
Suppose a and b are positive real numbers and are seminorms on X such that for every x \\in X, if then q(x) < b. Then
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

\nIf p is a seminorm on X and f is a linear functional on X then:\n

|f| \\leq p on X if and only if on X (see footnote for proof).\n
f \\leq p on X if and only if
If a > 0 and b > 0 are such that p(x) < a implies then for all x \\in X.

\n\n## Hahn–Banach theorem for seminorms\nSeminorms offer a particularly clean formulation of the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem):\nA similar extension property also holds for seminorms:\n\n# Topologies of seminormed spaces\n\n## Pseudometrics and the induced topology\nA seminorm p on X induces a topology, called the, via the canonical [translation-invariant](https://bliptext.com/articles/translation-invariant) [pseudometric](https://bliptext.com/articles/pseudometric-space) ;\nThis topology is [Hausdorff](https://bliptext.com/articles/hausdorff-space) if and only if d_p is a metric, which occurs if and only if p is a [norm](https://bliptext.com/articles/norm-mathematics).\nThis topology makes X into a [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) that has a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) neighborhood of the origin and a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) at the origin consisting of the following open balls (or the closed balls) centered at the origin:\nas r > 0 ranges over the positive reals.\nEvery seminormed space (X, p) should be assumed to be endowed with this topology unless indicated otherwise.\nA topological vector space whose topology is induced by some seminorm is called.\nEquivalently, every vector space X with seminorm p induces a [vector space quotient](https://bliptext.com/articles/quotient-space-linear-algebra) X / W, where W is the subspace of X consisting of all vectors x \\in X with p(x) = 0.\nThen X / W carries a norm defined by The resulting topology, [pulled back](https://bliptext.com/articles/pullback) to X, is precisely the topology induced by p.\nAny seminorm-induced topology makes X [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space), as follows.\nIf 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](https://bliptext.com/articles/convex-set) [balanced](https://bliptext.com/articles/balanced-set) sets that are open (resp. closed) in the p-topology on X.\n\n### Stronger, weaker, and equivalent seminorms\nThe notions of stronger and weaker semi[norms](https://bliptext.com/articles/norm-mathematics) are akin to the notions of stronger and weaker [norms](https://bliptext.com/articles/norm-mathematics).\nIf 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:\nThe seminorms p and q are called if they are both weaker (or both stronger) than each other.\nThis happens if they satisfy any of the following conditions:\n

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.
If is a sequence in X then if and only if
There exist positive real numbers r > 0 and R > 0 such that

\n\n## Normability and seminormability\nA topological vector space (TVS) is said to be a (respectively, a ) if its topology is induced by a single seminorm (resp. a single norm).\nA TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and [T1](https://bliptext.com/articles/t1-space) (because a TVS is Hausdorff if and only if it is a [T1](https://bliptext.com/articles/t1-space) space).\n****A**** **** ****i****s**** ****a**** ****t****o****p****o****l****o****g****i****c****a****l**** ****v****e****c****t****o****r**** ****s****p****a****c****e**** ****t****h****a****t**** ****p****o****s****s****e****s****s****e****s**** ****a**** ****b****o****u****n****d****e****d**** ****n****e****i****g****h****b****o****r****h****o****o****d**** ****o****f**** ****t****h****e**** ****o****r****i****g****i****n****.****\nNormability of [topological vector spaces](https://bliptext.com/articles/topological-vector-space) is characterized by [Kolmogorov's normability criterion](https://bliptext.com/articles/kolmogorov-s-normability-criterion).\nA TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.\nThus a [locally convex](https://bliptext.com/articles/locally-convex) TVS is seminormable if and only if it has a non-empty bounded open set.\nA TVS is normable if and only if it is a [T1 space](https://bliptext.com/articles/t1-space) and admits a bounded convex neighborhood of the origin.\nIf X is a Hausdorff [locally convex](https://bliptext.com/articles/locally-convex) TVS then the following are equivalent:\n

X is normable.
X is seminormable.
X has a bounded neighborhood of the origin.
The [strong dual](https://bliptext.com/articles/strong-dual) of X is normable.
The strong dual of X is [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space).

\nFurthermore, X is finite dimensional if and only if is normable (here denotes X^{\\prime} endowed with the [weak-* topology](https://bliptext.com/articles/weak-topology)).\nThe 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).\n\n## Topological properties\n

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 \\{0\\} in a locally convex space X whose topology is defined by a family of continuous seminorms \\mathcal{P} is equal to
A subset S in a seminormed space (X, p) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) if and only if p(S) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space).
If (X, p) is a seminormed space then the locally convex topology that p induces on X makes X into a [pseudometrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space) with a canonical pseudometric given by 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).

\n\n## Continuity of seminorms\nIf p is a seminorm on a topological vector space X, then the following are equivalent:\n

p is continuous.
p is continuous at 0;
is open in X;
is closed neighborhood of 0 in X;
p is uniformly continuous on X;
There exists a continuous seminorm q on X such that p \\leq q.

\nIn 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.\nIf 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.\n\n## Continuity of linear maps\nIf is a map between seminormed spaces then let\nIf is a linear map between seminormed spaces then the following are equivalent:\n

F is continuous;
;
There exists a real K \\geq 0 such that p \\leq K q;

\nIf F is continuous then for all x \\in X.\nThe space of all continuous linear maps between seminormed spaces is itself a seminormed space under the seminorm\nThis seminorm is a norm if q is a norm.\n\n# Generalizations\nThe concept of in [composition algebras](https://bliptext.com/articles/composition-algebra) does share the usual properties of a norm.\nA composition algebra (A, *, N) consists of an [algebra over a field](https://bliptext.com/articles/algebra-over-a-field) A, an [involution](https://bliptext.com/articles/involution-mathematics) \\,*, and a [quadratic form](https://bliptext.com/articles/quadratic-form) N, which is called the \"norm\".\nIn several cases N is an [isotropic quadratic form](https://bliptext.com/articles/isotropic-quadratic-form) so that A has at least one [null vector](https://bliptext.com/articles/null-vector), contrary to the separation of points required for the usual norm discussed in this article.\nAn or a is a seminorm that also satisfies\n**Weakening subadditivity: Quasi-seminorms**\nA map is called a if it is (absolutely) homogeneous and there exists some b \\leq 1 such that\nThe smallest value of b for which this holds is called the\nA quasi-seminorm that separates points is called a on X.\n**Weakening homogeneity - k-seminorms**\nA map is called a if it is subadditive and there exists a k such that and for all x \\in X and scalars s, A k-seminorm that separates points is called a on X.\nWe have the following relationship between quasi-seminorms and k-seminorms:","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:03:05.137Z","updated_at":"2025-01-25T19:03:05.137Z","search_vector":"'seminorm':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), particularly in [functional analysis](https://bliptext.com/articles/functional-analysis), a **semi[norm](https://bliptext.com/articles/norm-mathematics)** is a [norm](https://bliptext.com/articles/norm-mathematics) that need not be [positive definite](https://bliptext.com/articles/positive-definite).\nSeminorms are intimately connected with [convex sets](https://bliptext.com/articles/convex-set): every seminorm is the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) of some [absorbing](https://bliptext.com/articles/absorbing-set) [disk](https://bliptext.com/articles/absolutely-convex-set) and, conversely, the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) of any such set is a seminorm.\nA [topological vector space](https://bliptext.com/articles/topological-vector-space) is locally convex if and only if its topology is induced by a family of seminorms.\n\n# Definition\nLet X be a vector space over either the [real numbers](https://bliptext.com/articles/real-number) \\R or the [complex](https://bliptext.com/articles/complex-number) numbers \\Complex.\nA [real-valued function](https://bliptext.com/articles/real-valued-function) is called a if it satisfies the following two conditions:\nThese two conditions imply that p(0) = 0 and that every seminorm p also has the following property:\n [Nonnegativity](https://bliptext.com/articles/nonnegative): p(x) \\geq 0 for all x \\in X. \n \nSome 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.\nBy definition, a [norm](https://bliptext.com/articles/norm-mathematics) on X is a semi[norm](https://bliptext.com/articles/norm-mathematics) that also separates points, meaning that it has the following additional property:\n [Positive definite](https://bliptext.com/articles/positive-definite)/Positive/: whenever x \\in X satisfies p(x) = 0, then x = 0. \n \nA 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.\nSince absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a [sublinear function](https://bliptext.com/articles/sublinear-function).\nA map is called a if it is subadditive and [positive homogeneous](https://bliptext.com/articles/positive-homogeneous).\nUnlike a seminorm, a sublinear function is necessarily nonnegative.\nSublinear functions are often encountered in the context of the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem).\nA real-valued function is a seminorm if and only if it is a [sublinear](https://bliptext.com/articles/sublinear-function) and [balanced function](https://bliptext.com/articles/balanced-function).\n\n# Examples\n \n The on X, which refers to the constant 0 map on X, induces the [indiscrete topology](https://bliptext.com/articles/indiscrete-topology) on X. \n Let \\mu be a measure on a space \\Omega.\nFor an arbitrary constant c \\geq 1, let X be the set of all functions for which\nexists and is finite.\nIt can be shown that X is a vector space, and the functional is a seminorm on X. However, it is not always a norm (e.g. if and \\mu is the [Lebesgue measure](https://bliptext.com/articles/lebesgue-measure)) because does not always imply h = 0.\nTo make a norm, quotient X by the closed subspace of functions h with.\nThe [resulting space](https://bliptext.com/articles/lp-space), L^c(\\mu), has a norm induced by . \n If f is any [linear form](https://bliptext.com/articles/linear-form) on a vector space then its [absolute value](https://bliptext.com/articles/absolute-value) |f|, defined by is a seminorm. \n A [sublinear function](https://bliptext.com/articles/sublinear-function) on a real vector space X is a seminorm if and only if it is a, meaning that for all x \\in X. \n Every real-valued [sublinear function](https://bliptext.com/articles/sublinear-function) on a real vector space X induces a seminorm defined by \n Any finite sum of seminorms is a seminorm.\nThe restriction of a seminorm (respectively, norm) to a [vector subspace](https://bliptext.com/articles/vector-subspace) is once again a seminorm (respectively, norm). \n If and are seminorms (respectively, norms) on X and Y then the map defined by is a seminorm (respectively, a norm) on X \\times Y. In particular, the maps on X \\times Y defined by and are both seminorms on X \\times Y. \n If p and q are seminorms on X then so are\nand\nwhere and\n \n The space of seminorms on X is generally not a [distributive lattice](https://bliptext.com/articles/distributive-lattice) with respect to the above operations.\nFor example, over \\R^2, are such that\nwhile \n If L : X \\to Y is a [linear map](https://bliptext.com/articles/linear-map) and is a seminorm on Y, then 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 is a norm on L(X). \n \n\n# Minkowski functionals and seminorms\nSeminorms on a vector space X are intimately tied, via Minkowski functionals, to subsets of X that are [convex](https://bliptext.com/articles/convex-set), [balanced](https://bliptext.com/articles/balanced-set), and [absorbing](https://bliptext.com/articles/absorbing-set).\nGiven such a subset D of X, the Minkowski functional of D is a seminorm.\nConversely, 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.\n\n# Algebraic properties\nEvery seminorm is a [sublinear function](https://bliptext.com/articles/sublinear-function), and thus satisfies all properties of a [sublinear function](https://bliptext.com/articles/sublinear-function), including [convexity](https://bliptext.com/articles/convex-function), p(0) = 0, and for all vectors x, y \\in X: the [reverse triangle inequality](https://bliptext.com/articles/reverse-triangle-inequality):\nand also\nand\nFor any vector x \\in X and positive real r > 0:\nand furthermore, is an [absorbing](https://bliptext.com/articles/absorbing-set) [disk](https://bliptext.com/articles/absolutely-convex-set) in X.\nIf 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 and furthermore, for any linear functional g on X, g \\leq p on X if and only if\n**Other properties of seminorms**\nEvery seminorm is a [balanced function](https://bliptext.com/articles/balanced-function).\nA seminorm p is a norm on X if and only if does not contain a non-trivial vector subspace.\nIf is a seminorm on X then is a vector subspace of X and for every x \\in X, p is constant on the set and equal to p(x).\nFurthermore, for any real r > 0,\nIf D is a set satisfying then D is [absorbing](https://bliptext.com/articles/absorbing-set) in X and p = p_D where p_D denotes the [Minkowski functional](https://bliptext.com/articles/minkowski-functional) associated with D (that is, the gauge of D).\nIn particular, if D is as above and q is any seminorm on X, then q = p if and only if\nIf is a normed space and x, y \\in X then for all z in the interval [x, y].\nEvery norm is a [convex function](https://bliptext.com/articles/convex-function) and consequently, finding a global maximum of a norm-based [objective function](https://bliptext.com/articles/objective-function) is sometimes tractable.\n\n## Relationship to other norm-like concepts\nLet be a non-negative function.\nThe following are equivalent:\n \n p is a seminorm. \n p is a [convex](https://bliptext.com/articles/convex-function) [F-seminorm](https://bliptext.com/articles/f-seminorm). \n p is a convex balanced G-seminorm. \n \nIf any of the above conditions hold, then the following are equivalent:\n \n p is a norm; \n does not contain a non-trivial vector subspace. \n There exists a [norm](https://bliptext.com/articles/normed-vector-space) on X, with respect to which, is bounded. \n \nIf p is a sublinear function on a real vector space X then the following are equivalent:\n \n p is a [linear functional](https://bliptext.com/articles/linear-functional); \n ; \n ; \n \n\n## Inequalities involving seminorms\nIf are seminorms on X then:\n \n p \\leq q if and only if q(x) \\leq 1 implies \n If a > 0 and b > 0 are such that p(x) \n Suppose a and b are positive real numbers and are seminorms on X such that for every x \\in X, if then q(x) \n 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 \n \nIf p is a seminorm on X and f is a linear functional on X then:\n \n |f| \\leq p on X if and only if on X (see footnote for proof).\n \n f \\leq p on X if and only if \n If a > 0 and b > 0 are such that p(x) \n \n\n## Hahn–Banach theorem for seminorms\nSeminorms offer a particularly clean formulation of the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem):\nA similar extension property also holds for seminorms:\n\n# Topologies of seminormed spaces\n\n## Pseudometrics and the induced topology\nA seminorm p on X induces a topology, called the, via the canonical [translation-invariant](https://bliptext.com/articles/translation-invariant) [pseudometric](https://bliptext.com/articles/pseudometric-space) ;\nThis topology is [Hausdorff](https://bliptext.com/articles/hausdorff-space) if and only if d_p is a metric, which occurs if and only if p is a [norm](https://bliptext.com/articles/norm-mathematics).\nThis topology makes X into a [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space) [topological vector space](https://bliptext.com/articles/topological-vector-space) that has a [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) neighborhood of the origin and a [neighborhood basis](https://bliptext.com/articles/neighborhood-basis) at the origin consisting of the following open balls (or the closed balls) centered at the origin:\nas r > 0 ranges over the positive reals.\nEvery seminormed space (X, p) should be assumed to be endowed with this topology unless indicated otherwise.\nA topological vector space whose topology is induced by some seminorm is called.\nEquivalently, every vector space X with seminorm p induces a [vector space quotient](https://bliptext.com/articles/quotient-space-linear-algebra) X / W, where W is the subspace of X consisting of all vectors x \\in X with p(x) = 0.\nThen X / W carries a norm defined by The resulting topology, [pulled back](https://bliptext.com/articles/pullback) to X, is precisely the topology induced by p.\nAny seminorm-induced topology makes X [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space), as follows.\nIf 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](https://bliptext.com/articles/convex-set) [balanced](https://bliptext.com/articles/balanced-set) sets that are open (resp. closed) in the p-topology on X.\n\n### Stronger, weaker, and equivalent seminorms\nThe notions of stronger and weaker semi[norms](https://bliptext.com/articles/norm-mathematics) are akin to the notions of stronger and weaker [norms](https://bliptext.com/articles/norm-mathematics).\nIf 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:\nThe seminorms p and q are called if they are both weaker (or both stronger) than each other.\nThis happens if they satisfy any of the following conditions:\n \n The topology on X induced by q is the same as the topology induced by p. \n q is stronger than p and p is stronger than q. \n If is a sequence in X then if and only if \n There exist positive real numbers r > 0 and R > 0 such that \n \n\n## Normability and seminormability\nA topological vector space (TVS) is said to be a (respectively, a ) if its topology is induced by a single seminorm (resp. a single norm).\nA TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and [T1](https://bliptext.com/articles/t1-space) (because a TVS is Hausdorff if and only if it is a [T1](https://bliptext.com/articles/t1-space) space).\n****A**** **** ****i****s**** ****a**** ****t****o****p****o****l****o****g****i****c****a****l**** ****v****e****c****t****o****r**** ****s****p****a****c****e**** ****t****h****a****t**** ****p****o****s****s****e****s****s****e****s**** ****a**** ****b****o****u****n****d****e****d**** ****n****e****i****g****h****b****o****r****h****o****o****d**** ****o****f**** ****t****h****e**** ****o****r****i****g****i****n****.****\nNormability of [topological vector spaces](https://bliptext.com/articles/topological-vector-space) is characterized by [Kolmogorov's normability criterion](https://bliptext.com/articles/kolmogorov-s-normability-criterion).\nA TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.\nThus a [locally convex](https://bliptext.com/articles/locally-convex) TVS is seminormable if and only if it has a non-empty bounded open set.\nA TVS is normable if and only if it is a [T1 space](https://bliptext.com/articles/t1-space) and admits a bounded convex neighborhood of the origin.\nIf X is a Hausdorff [locally convex](https://bliptext.com/articles/locally-convex) TVS then the following are equivalent:\n \n X is normable. \n X is seminormable. \n X has a bounded neighborhood of the origin. \n The [strong dual](https://bliptext.com/articles/strong-dual) of X is normable. \n The strong dual of X is [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space). \n \nFurthermore, X is finite dimensional if and only if is normable (here denotes X^{\\prime} endowed with the [weak-* topology](https://bliptext.com/articles/weak-topology)).\nThe 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).\n\n## Topological properties\n \n If X is a TVS and p is a continuous seminorm on X, then the closure of in X is equal to \n 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 \n A subset S in a seminormed space (X, p) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space) if and only if p(S) is [bounded](https://bliptext.com/articles/bounded-set-topological-vector-space). \n If (X, p) is a seminormed space then the locally convex topology that p induces on X makes X into a [pseudometrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space) with a canonical pseudometric given by for all x, y \\in X. \n 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). \n \n\n## Continuity of seminorms\nIf p is a seminorm on a topological vector space X, then the following are equivalent:\n \n p is continuous. \n p is continuous at 0; \n is open in X; \n is closed neighborhood of 0 in X; \n p is uniformly continuous on X; \n There exists a continuous seminorm q on X such that p \\leq q. \n \nIn 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.\nIf 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.\n\n## Continuity of linear maps\nIf is a map between seminormed spaces then let\nIf is a linear map between seminormed spaces then the following are equivalent:\n \n F is continuous; \n ; \n There exists a real K \\geq 0 such that p \\leq K q; \nIf F is continuous then for all x \\in X.\nThe space of all continuous linear maps between seminormed spaces is itself a seminormed space under the seminorm\nThis seminorm is a norm if q is a norm.\n\n# Generalizations\nThe concept of in [composition algebras](https://bliptext.com/articles/composition-algebra) does share the usual properties of a norm.\nA composition algebra (A, *, N) consists of an [algebra over a field](https://bliptext.com/articles/algebra-over-a-field) A, an [involution](https://bliptext.com/articles/involution-mathematics) \\,*, and a [quadratic form](https://bliptext.com/articles/quadratic-form) N, which is called the \"norm\".\nIn several cases N is an [isotropic quadratic form](https://bliptext.com/articles/isotropic-quadratic-form) so that A has at least one [null vector](https://bliptext.com/articles/null-vector), contrary to the separation of points required for the usual norm discussed in this article.\nAn or a is a seminorm that also satisfies\n**Weakening subadditivity: Quasi-seminorms**\nA map is called a if it is (absolutely) homogeneous and there exists some b \\leq 1 such that\nThe smallest value of b for which this holds is called the\nA quasi-seminorm that separates points is called a on X.\n**Weakening homogeneity - k-seminorms**\nA map is called a if it is subadditive and there exists a k such that and for all x \\in X and scalars s, A k-seminorm that separates points is called a on X.\nWe have the following relationship between quasi-seminorms and k-seminorms:","revisionCount":0},"hasMore":false,"nextCursor":null}

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