Math 2.0

Functional analysis Definition

Equivalent norms

Builds on Equivalent metricsMetric topology

Definition

Let VV be a vector space over ℝ\mathbb{R} or ℂ\mathbb{C}. A norm on VV is a function ∥⋅∥ ⁣:V→[0,∞)\|\cdot\| \colon V \to [0, \infty) such that, for all x,y∈Vx, y \in V and all scalars λ\lambda: ∥x∥=0\|x\| = 0 only for x=0x = 0, ∥λx∥=∣λ∣ ∥x∥\|\lambda x\| = |\lambda| \, \|x\|, and ∥x+y∥≤∥x∥+∥y∥\|x + y\| \le \|x\| + \|y\|. Two norms ∥⋅∥a\|\cdot\|_a and ∥⋅∥b\|\cdot\|_b on VV are equivalent when there are real numbers c>0c > 0 and C>0C > 0 with c ∥x∥a≤∥x∥b≤C ∥x∥afor all x∈V.c \, \|x\|_a \le \|x\|_b \le C \, \|x\|_a \quad \text{for all } x \in V.

Remarks

Each norm gives a metric d(x,y)=∥x−y∥d(x, y) = \|x - y\|. Two norms are equivalent if and only if their metrics are strongly equivalent. For norms, equivalence is the same as giving the same metric topology. On a finite-dimensional space, all norms are equivalent. On C([0,1])C([0, 1]), the sup norm and the norm ∫01∣f(t)∣ dt\int_0^1 |f(t)| \, dt are not equivalent.

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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 relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, 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 metricsdefinitionEquivalent norms, definition, no evidence levelEquivalent normsdefinition
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  1. Element definition
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  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. Topological space definition
  22. Function definition
  23. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Equivalent metrics definition
  28. Equivalent norms definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Norm_(mathematics), Definition
  2. Web https://en.wikipedia.org/wiki/Norm_(mathematics), Equivalence
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DefinesEquivalent norms
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