{"history":[],"article":{"id":"0a54c0ca-ab28-4acd-8822-1cc68d919703","title":"F-space","slug":"f-space","content":":::summary\n:::\n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis), an **F-space** is a [vector space](https://bliptext.com/articles/vector-space) X over the [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) numbers together with a [metric](https://bliptext.com/articles/metric-mathematics) such that\nThe operation is called an **F-norm**, although in general an **F-norm** is not required to be homogeneous.\nBy [translation-invariance](https://bliptext.com/articles/translation-invariance), the metric is recoverable from the F-norm.\nThus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm.\nSome authors use the term rather than, but usually the term \"Fréchet space\" is reserved for [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) F-spaces.\nSome other authors use the term \"F-space\" as a synonym of \"Fréchet space\", by which they mean a locally convex complete metrizable [topological vector space](https://bliptext.com/articles/topological-vector-space).\nThe metric may or may not necessarily be part of the structure on an F-space; many authors only require that such a space be [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) in a manner that satisfies the above properties.\n\n# Examples\nAll [Banach spaces](https://bliptext.com/articles/banach-space) and [Fréchet spaces](https://bliptext.com/articles/fr-chet-space) are F-spaces.\nIn particular, a Banach space is an F-space with an additional requirement that\nThe [Lp spaces](https://bliptext.com/articles/lp-space) can be made into F-spaces for all p \\geq 0 and for p \\geq 1 they can be made into locally convex and thus Fréchet spaces and even Banach spaces.\n\n## Example 1\nis an F-space.\nIt admits no continuous seminorms and no continuous linear functionals — it has trivial [dual space](https://bliptext.com/articles/dual-space).\n\n## Example 2\nLet be the space of all complex valued [Taylor series](https://bliptext.com/articles/taylor-series)\non the unit disc \\mathbb{D} such that\nthen for 0 < p < 1, are F-spaces under the [p-norm](https://bliptext.com/articles/p-norm):\nIn fact, W_p is a [quasi-Banach algebra](https://bliptext.com/articles/quasi-banach-algebra).\nMoreover, for any \\zeta with the map is a bounded linear (multiplicative functional) on\n\n# Sufficient conditions\n\n# Related properties\nThe [open mapping theorem](https://bliptext.com/articles/open-mapping-theorem-functional-analysis) implies that if are topologies on X that make both (X, \\tau) and into [complete](https://bliptext.com/articles/complete-topological-vector-space) [metrizable topological vector spaces](https://bliptext.com/articles/metrizable-topological-vector-space) (for example, Banach or [Fréchet spaces](https://bliptext.com/articles/fr-chet-space)) and if one topology is [finer or coarser](https://bliptext.com/articles/comparison-of-topologies) than the other then they must be equal (that is, if ).\n

A linear [almost continuous](https://bliptext.com/articles/almost-continuous) map into an F-space whose graph is closed is continuous.
A linear [almost open](https://bliptext.com/articles/almost-open-map) map into an F-space whose graph is closed is necessarily an [open map](https://bliptext.com/articles/open-map).
A linear continuous [almost open](https://bliptext.com/articles/almost-open-map) map from an F-space is necessarily an [open map](https://bliptext.com/articles/open-map).
A linear continuous almost [open map](https://bliptext.com/articles/open-map) from an F-space whose image is of the [second category](https://bliptext.com/articles/second-category) in the codomain is necessarily a [surjective](https://bliptext.com/articles/surjection) [open map](https://bliptext.com/articles/open-map).

\n\n# Sources\n[Przestrzeń Frécheta (analiza funkcjonalna)](https://bliptext.com/articles/przestrze-fr-cheta-analiza-funkcjonalna)","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:39:12.065Z","updated_at":"2025-01-25T18:39:12.065Z","search_vector":"'f':2 'f-space':1 'space':3","search_content":" \n \n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis), an **F-space** is a [vector space](https://bliptext.com/articles/vector-space) X over the [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) numbers together with a [metric](https://bliptext.com/articles/metric-mathematics) such that\nThe operation is called an **F-norm**, although in general an **F-norm** is not required to be homogeneous.\nBy [translation-invariance](https://bliptext.com/articles/translation-invariance), the metric is recoverable from the F-norm.\nThus, a real or complex F-space is equivalently a real or complex vector space equipped with a complete F-norm.\nSome authors use the term rather than, but usually the term \"Fréchet space\" is reserved for [locally convex](https://bliptext.com/articles/locally-convex-topological-vector-space) F-spaces.\nSome other authors use the term \"F-space\" as a synonym of \"Fréchet space\", by which they mean a locally convex complete metrizable [topological vector space](https://bliptext.com/articles/topological-vector-space).\nThe metric may or may not necessarily be part of the structure on an F-space; many authors only require that such a space be [metrizable](https://bliptext.com/articles/metrizable-topological-vector-space) in a manner that satisfies the above properties.\n\n# Examples\nAll [Banach spaces](https://bliptext.com/articles/banach-space) and [Fréchet spaces](https://bliptext.com/articles/fr-chet-space) are F-spaces.\nIn particular, a Banach space is an F-space with an additional requirement that\nThe [Lp spaces](https://bliptext.com/articles/lp-space) can be made into F-spaces for all p \\geq 0 and for p \\geq 1 they can be made into locally convex and thus Fréchet spaces and even Banach spaces.\n\n## Example 1\nis an F-space.\nIt admits no continuous seminorms and no continuous linear functionals — it has trivial [dual space](https://bliptext.com/articles/dual-space).\n\n## Example 2\nLet be the space of all complex valued [Taylor series](https://bliptext.com/articles/taylor-series)\non the unit disc \\mathbb{D} such that\nthen for 0 \n A linear [almost continuous](https://bliptext.com/articles/almost-continuous) map into an F-space whose graph is closed is continuous. \n A linear [almost open](https://bliptext.com/articles/almost-open-map) map into an F-space whose graph is closed is necessarily an [open map](https://bliptext.com/articles/open-map). \n A linear continuous [almost open](https://bliptext.com/articles/almost-open-map) map from an F-space is necessarily an [open map](https://bliptext.com/articles/open-map). \n A linear continuous almost [open map](https://bliptext.com/articles/open-map) from an F-space whose image is of the [second category](https://bliptext.com/articles/second-category) in the codomain is necessarily a [surjective](https://bliptext.com/articles/surjection) [open map](https://bliptext.com/articles/open-map). \n \n\n# Sources\n[Przestrzeń Frécheta (analiza funkcjonalna)](https://bliptext.com/articles/przestrze-fr-cheta-analiza-funkcjonalna)","revisionCount":0},"hasMore":false,"nextCursor":null}

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