Math 2.0

Functional analysis Example

Sequence spaces lp

Builds on Banach spacep-norm on R^n

Example

Let p≥1p \ge 1 be a real number. Let ℓp\ell^p be the set of real sequences x=(xn)n≥1x = (x_n)_{n \ge 1} with ∑n≥1∣xn∣p<∞\sum_{n \ge 1} |x_n|^p < \infty, and let ∥x∥p=(∑n≥1∣xn∣p)1/p.\|x\|_p = \left( \sum_{n \ge 1} |x_n|^p \right)^{1/p}. Let ℓ∞\ell^\infty be the set of bounded real sequences, with ∥x∥∞=sup⁡n≥1∣xn∣\|x\|_\infty = \sup_{n \ge 1} |x_n|. Then each ℓp\ell^p, for 1≤p≤∞1 \le p \le \infty, is a vector space, ∥⋅∥p\|\cdot\|_p is a norm on it, and ℓp\ell^p is a Banach space.

Remarks

The norms extend the p-norms on R^n to infinite sequences. For 1≤p<q≤∞1 \le p < q \le \infty, we have ℓp⊆ℓq\ell^p \subseteq \ell^q, and the inclusion is strict. For example, (1/n)(1/n) is in ℓ2\ell^2 but not in ℓ1\ell^1. The space ℓ2\ell^2 has the inner product ⟨x,y⟩=∑nxnyn\langle x, y \rangle = \sum_n x_n y_n, and it is a Hilbert space. The closed unit ball of ℓp\ell^p is not compact: the unit vectors eke_k satisfy ∥ej−ek∥p=21/p\|e_j - e_k\|_p = 2^{1/p} for j≠kj \neq k and p<∞p < \infty.

Reading path

To understand this page from the ground up, read these 39 pages in order. The foundations come first.

Show the path as a graph
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 abovedefinitionp-metric on R^n, example, no evidence levelp-metric on R^nexampleSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionSupremum, definition, no evidence levelSupremumdefinitionp-norm on R^n, example, no evidence levelp-norm on R^nexampleEmpty 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 spacedefinitionSequence spaces lp, example, no evidence levelSequence spaces lpexample
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. p-metric on R^n example
  7. Set definition
  8. Conjunction definition
  9. Disjunction definition
  10. Implication definition
  11. Predicate definition
  12. Supremum definition
  13. p-norm on R^n example
  14. Empty set object
  15. Family of sets definition
  16. Ordered pair definition
  17. Intersection definition
  18. Union definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. Union of a family of sets definition
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Topological space definition
  27. Function definition
  28. Metric space definition
  29. Convergence of a sequence in a metric space definition
  30. Open ball definition
  31. Diameter of a set definition
  32. Metric topology definition
  33. Epsilon characterisation of Cauchy sequences statement
  34. Complete metric space definition
  35. Open set definition
  36. Equivalent metrics definition
  37. Equivalent norms definition
  38. Banach space definition
  39. Sequence spaces lp example

Where this leads

12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sequence_space, ℓp spaces
  2. Web https://en.wikipedia.org/wiki/Lp_space, ℓp spaces and sequence spaces
About this page
Tag08P6T2WH6R
URIm2://08P6T2WH6R
Aliaseslp-sequence-space
Kindexample
WritingW2 usable
MSC46B45
Instance ofBanach space
UsesBanach space, p-norm on R^n

Markdown · JSON · Edit on GitHub

What this page needs for W3

  • a person confirmed each agent-written section: rule P04 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by