Math 2.0

Functional analysis Normed and Banach spaces Lemma

Riesz's lemma

Builds on Closed setEquivalent norms

Statement

Let XX be a normed space over ℝ\mathbb{R} or ℂ\mathbb{C}, and let Y≠XY \neq X be a closed linear subspace of XX. Let α\alpha be a real number with 0<α<10 < \alpha < 1. Then there is x∈Xx \in X with ∥x∥=1\|x\| = 1 and ∥x−y∥≥α\|x - y\| \ge \alpha for all y∈Yy \in Y.

Remarks

Proof idea: take x0∉Yx_0 \notin Y and let r=inf⁡y∈Y∥x0−y∥>0r = \inf_{y \in Y} \|x_0 - y\| > 0, which is positive because YY is closed. Choose y0∈Yy_0 \in Y with ∥x0−y0∥≤r/α\|x_0 - y_0\| \le r / \alpha, and let x=(x0−y0)/∥x0−y0∥x = (x_0 - y_0) / \|x_0 - y_0\|. If YY is finite-dimensional, one can take α=1\alpha = 1. The lemma is a substitute for orthogonal vectors in normed spaces without an inner product. Its main application is that a compact closed unit ball forces finite dimension.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty 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 relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionEquivalent metrics, definition, no evidence levelEquivalent metricsdefinitionClosed set, definition, no evidence levelClosed setdefinitionEquivalent norms, definition, no evidence levelEquivalent normsdefinitionRiesz's lemma, statement, no evidence levelRiesz's lemmastatement
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. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Equivalent metrics definition
  30. Closed set definition
  31. Equivalent norms definition
  32. Riesz's lemma statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Riesz%27s_lemma, Statement
About this page
Tag097C7KCAT5
URIm2://097C7KCAT5
Aliasesriesz-lemma
Kindstatement, lemma
WritingW0 stub
MSC46B20
StatusThis copy of the site has no build data. Run m2 status 097C7KCAT5.
UsesClosed set, Equivalent norms
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