{"history":[],"article":{"id":"f4a35951-dc1b-4911-8edb-cc0ca7bad1c8","title":"Quasinorm","slug":"quasinorm","content":":::summary\n:::\n\n# \nIn [linear algebra](https://bliptext.com/articles/linear-algebra), [functional analysis](https://bliptext.com/articles/functional-analysis) and related areas of [mathematics](https://bliptext.com/articles/mathematics), a **quasi[norm](https://bliptext.com/articles/norm-mathematics)** is similar to a [norm](https://bliptext.com/articles/norm-mathematics) in that it satisfies the [norm](https://bliptext.com/articles/norm-mathematics) axioms, except that the [triangle inequality](https://bliptext.com/articles/triangle-inequality) is replaced by\nfor some K > 1.\n\n# Definition\nA on a vector space X is a real-valued map p on X that satisfies the following conditions:\n

: p \\geq 0;
: for all x \\in X and all scalars s;
there exists a real k \\geq 1 such that for all x, y \\in X.

\nA is a quasi-seminorm that also satisfies:\n

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

\n\nA pair (X, p) consisting of a [vector space](https://bliptext.com/articles/vector-space) X and an associated quasi-seminorm p is called a.\nIf the quasi-seminorm is a quasinorm then it is also called a.\n**Multiplier**\nThe [infimum](https://bliptext.com/articles/infimum) of all values of k that satisfy condition (3) is called the of p. The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition.\nThe term is sometimes used to describe a quasi-seminorm whose multiplier is equal to k.\nA (respectively, a ) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is 1.\nThus every [seminorm](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) is a quasi-[seminorm](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) and every [norm](https://bliptext.com/articles/norm-mathematics) is a quasi[norm](https://bliptext.com/articles/norm-mathematics) (and a quasi-[seminorm](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics))).\n\n## Topology\nIf p is a quasinorm on X then p induces a vector topology on X whose neighborhood basis at the origin is given by the sets:\nas n ranges over the positive integers.\nA [topological vector space](https://bliptext.com/articles/topological-vector-space) with such a topology is called a or just a.\nEvery quasinormed topological vector space is [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space).\nA [complete](https://bliptext.com/articles/complete-topological-vector-space) quasinormed space is called a.\nEvery [Banach space](https://bliptext.com/articles/banach-space) is a quasi-[Banach space](https://bliptext.com/articles/banach-space), although not conversely.\n\n## Related definitions\nA quasinormed space is called a if the vector space A is an [algebra](https://bliptext.com/articles/algebra-over-a-field) and there is a constant K > 0 such that\nfor all x, y \\in A.\nA [complete](https://bliptext.com/articles/complete-topological-vector-space) quasinormed algebra is called a.\n\n# Characterizations\nA [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.\n\n# Examples\nSince every norm is a quasinorm, every [normed space](https://bliptext.com/articles/normed-space) is also a quasi[normed space](https://bliptext.com/articles/normed-space).\n**L^p spaces with 0 < p < 1**\nThe [L^p spaces](https://bliptext.com/articles/lp-space) for 0 < p < 1 are quasinormed spaces (indeed, they are even [F-spaces](https://bliptext.com/articles/f-space)) but they are not, in general, [normable](https://bliptext.com/articles/normable-space) (meaning that there might not exist any norm that defines their topology).\nFor 0 < p < 1, the [Lebesgue space](https://bliptext.com/articles/lp-space) L^p([0, 1]) is a [complete](https://bliptext.com/articles/complete-metric-space) [metrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space) (an [F-space](https://bliptext.com/articles/f-space)) that is [locally convex](https://bliptext.com/articles/locally-[convex](https://bliptext.com/articles/convex-set)-topological-vector-space) (in fact, its only [convex](https://bliptext.com/articles/convex-set) open subsets are itself L^p([0, 1]) and the empty set) and the [continuous linear functional](https://bliptext.com/articles/continuous-linear-functional) on L^p([0, 1]) is the constant 0 function.\nIn particular, the [Hahn-Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) does hold for L^p([0, 1]) when 0 < p < 1.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T08:19:16.828Z","updated_at":"2025-01-26T08:19:16.828Z","search_vector":"'quasinorm':1","search_content":" \n \n\n# \nIn [linear algebra](https://bliptext.com/articles/linear-algebra), [functional analysis](https://bliptext.com/articles/functional-analysis) and related areas of [mathematics](https://bliptext.com/articles/mathematics), a **quasi[norm](https://bliptext.com/articles/norm-mathematics)** is similar to a [norm](https://bliptext.com/articles/norm-mathematics) in that it satisfies the [norm](https://bliptext.com/articles/norm-mathematics) axioms, except that the [triangle inequality](https://bliptext.com/articles/triangle-inequality) is replaced by\nfor some K > 1.\n\n# Definition\nA on a vector space X is a real-valued map p on X that satisfies the following conditions:\n \n : p \\geq 0; \n : for all x \\in X and all scalars s; \n there exists a real k \\geq 1 such that for all x, y \\in X. \nA is a quasi-seminorm that also satisfies:\n [Positive definite](https://bliptext.com/articles/positive-definite)/: if x \\in X satisfies p(x) = 0, then x = 0. \n \nA pair (X, p) consisting of a [vector space](https://bliptext.com/articles/vector-space) X and an associated quasi-seminorm p is called a.\nIf the quasi-seminorm is a quasinorm then it is also called a.\n**Multiplier**\nThe [infimum](https://bliptext.com/articles/infimum) of all values of k that satisfy condition (3) is called the of p. The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition.\nThe term is sometimes used to describe a quasi-seminorm whose multiplier is equal to k.\nA (respectively, a ) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is 1.\nThus every [semi[norm](https://bliptext.com/articles/norm-mathematics)](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) is a quasi-[semi[norm](https://bliptext.com/articles/norm-mathematics)](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics)) and every [norm](https://bliptext.com/articles/norm-mathematics) is a quasi[norm](https://bliptext.com/articles/norm-mathematics) (and a quasi-[semi[norm](https://bliptext.com/articles/norm-mathematics)](https://bliptext.com/articles/semi[norm](https://bliptext.com/articles/norm-mathematics))).\n\n## Topology\nIf p is a quasinorm on X then p induces a vector topology on X whose neighborhood basis at the origin is given by the sets:\nas n ranges over the positive integers.\nA [topological vector space](https://bliptext.com/articles/topological-vector-space) with such a topology is called a or just a.\nEvery quasinormed topological vector space is [pseudometrizable](https://bliptext.com/articles/metrizable-topological-vector-space).\nA [complete](https://bliptext.com/articles/complete-topological-vector-space) quasinormed space is called a.\nEvery [Banach space](https://bliptext.com/articles/banach-space) is a quasi-[Banach space](https://bliptext.com/articles/banach-space), although not conversely.\n\n## Related definitions\nA quasinormed space is called a if the vector space A is an [algebra](https://bliptext.com/articles/algebra-over-a-field) and there is a constant K > 0 such that\nfor all x, y \\in A.\nA [complete](https://bliptext.com/articles/complete-topological-vector-space) quasinormed algebra is called a.\n\n# Characterizations\nA [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.\n\n# Examples\nSince every norm is a quasinorm, every [normed space](https://bliptext.com/articles/normed-space) is also a quasi[normed space](https://bliptext.com/articles/normed-space).\n**L^p spaces with 0 < p < 1**\nThe [L^p spaces](https://bliptext.com/articles/lp-space) for 0 < p < 1 are quasinormed spaces (indeed, they are even [F-spaces](https://bliptext.com/articles/f-space)) but they are not, in general, [normable](https://bliptext.com/articles/normable-space) (meaning that there might not exist any norm that defines their topology).\nFor 0 < p < 1, the [Lebesgue space](https://bliptext.com/articles/lp-space) L^p([0, 1]) is a [complete](https://bliptext.com/articles/complete-metric-space) [metrizable TVS](https://bliptext.com/articles/metrizable-topological-vector-space) (an [F-space](https://bliptext.com/articles/f-space)) that is [locally [convex](https://bliptext.com/articles/convex-set)](https://bliptext.com/articles/locally-[convex](https://bliptext.com/articles/convex-set)-topological-vector-space) (in fact, its only [convex](https://bliptext.com/articles/convex-set) open subsets are itself L^p([0, 1]) and the empty set) and the [continuous linear functional](https://bliptext.com/articles/continuous-linear-functional) on L^p([0, 1]) is the constant 0 function.\nIn particular, the [Hahn-Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) does hold for L^p([0, 1]) when 0 < p < 1.","revisionCount":0},"hasMore":false,"nextCursor":null}

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