Math 2.0

Functional analysis Definition

F-space

Builds on Metric spaceComplete metric space

Definition

Let 𝕂\mathbb{K} be ℝ\R or ℂ\C, and let EE be a topological vector space over 𝕂\mathbb{K}. EE is an F-space if there is a metric dd on EE such that:

  1. dd induces the topology of EE.
  2. dd is translation-invariant: d(x+z,y+z)=d(x,y)d(x + z, y + z) = d(x, y) for all x,y,z∈Ex, y, z \in E.
  3. The metric space (E,d)(E, d) is complete.

Remarks

Some authors use the name for a vector space with a metric satisfying 2 and 3 for which addition and scalar multiplication are continuous; this is the same notion. Example: every Banach space is an F-space with d(x,y)=∥x−y∥d(x, y) = \norm{x - y}; this metric induces the norm topology, it is translation-invariant since (x+z)−(y+z)=x−y(x + z) - (y + z) = x - y, and it is complete by definition of a Banach space. Example: for 0<p<10 < p < 1, Lp([0,1])L^p([0, 1]) with d(f,g)=∫01∣f−g∣pd(f, g) = \int_0^1 \abs{f - g}^p is an F-space that is not locally convex. Non-example: the space c00c_{00} of finitely supported real sequences with d(x,y)=sup⁡n∣xn−yn∣d(x, y) = \sup_n \abs{x_n - y_n} satisfies 1 and 2 but not 3: the sequences x(k)=(1,1/2,…,1/k,0,0,…)x^{(k)} = (1, 1/2, \ldots, 1/k, 0, 0, \ldots) are Cauchy and have no limit in c00c_{00}. Every metrizable topological vector space has a translation-invariant metric inducing its topology, so condition 2 matters only together with condition 3.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionExtended real number line, object, no evidence levelExtended real number …objectProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statementComplete metric space, definition, no evidence levelComplete metric spacedefinitionF-space, definition, no evidence levelF-spacedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Element definition
  2. Extended real number line object
  3. Proposition concept
  4. Real numbers object
  5. Set bounded above definition
  6. Set definition
  7. Implication definition
  8. Predicate definition
  9. Supremum definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Metric space definition
  19. Convergence of a sequence in a metric space definition
  20. Diameter of a set definition
  21. Epsilon characterisation of Cauchy sequences statement
  22. Complete metric space definition
  23. F-space definition

Further reading

  1. Web https://en.wikipedia.org/wiki/F-space
About this page
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WritingW0 stub
MSC46A16
DefinesF-space
UsesMetric space, Complete metric space
Changed2 November 2026

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