Math 2.0

Functional analysis Definition

Banach space

Builds on Equivalent normsComplete metric space

Definition

Let VV be a vector space over ℝ\mathbb{R} or ℂ\mathbb{C} with a norm ∥⋅∥\|\cdot\|, and let d(x,y)=∥x−y∥d(x, y) = \|x - y\|. The space (V,∥⋅∥)(V, \|\cdot\|) is a Banach space when the metric space (V,d)(V, d) is complete.

Remarks

Examples are ℝn\mathbb{R}^n with any norm, the spaces ell p for 1≤p≤∞1 \le p \le \infty, and Cb(X)C_b(X) with the sup norm. The space of polynomial functions on [0,1][0, 1] with the sup norm is not a Banach space. Its completion is C([0,1])C([0, 1]). In a normed space VV, completeness is the same as: each absolutely convergent series converges. Each finite-dimensional normed space is complete.

Reading path

To understand this page from the ground up, read these 36 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 levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionOpen ball, definition, no evidence levelOpen balldefinitionDiameter of a set, definition, no evidence levelDiameter of a setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionEpsilon characterisation of Cauchy sequences, statement, no evidence levelEpsilon characterisat…statementComplete metric space, definition, no evidence levelComplete metric spacedefinitionOpen set, definition, no evidence levelOpen setdefinitionEquivalent metrics, definition, no evidence levelEquivalent metricsdefinitionEquivalent norms, definition, no evidence levelEquivalent normsdefinitionBanach space, definition, no evidence levelBanach 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.
  • no evidence level
  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. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Supremum definition
  12. Empty set object
  13. Family of sets definition
  14. Ordered pair definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Topological space definition
  25. Function definition
  26. Metric space definition
  27. Convergence of a sequence in a metric space definition
  28. Open ball definition
  29. Diameter of a set definition
  30. Metric topology definition
  31. Epsilon characterisation of Cauchy sequences statement
  32. Complete metric space definition
  33. Open set definition
  34. Equivalent metrics definition
  35. Equivalent norms definition
  36. Banach space definition

Where this leads

36 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Banach_space, Definition
About this page
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MSC46B99
DefinesBanach space
UsesEquivalent norms, Complete metric space

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