About: Bounded operator

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Linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X

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dbo:description
  • transformación lineal L entre espacios vectoriales normados X e Y para los cuales la relación entre la norma de L(v) y la de v está acotada por el mismo número, sobre todos los vectores distintos de cero v en X (es)
  • lineara operatoro inter normitaj spacoj, kies operatora normo estas finia (eo)
  • linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X (en)
dbo:wikiPageWikiLink
dbp:id
  • p/b017420 (en)
dbp:proof
  • Suppose that is bounded. Then, for all vectors with nonzero we have Letting go to zero shows that is continuous at Moreover, since the constant does not depend on this shows that in fact is uniformly continuous, and even Lipschitz continuous. Conversely, it follows from the continuity at the zero vector that there exists a such that for all vectors with Thus, for all non-zero one has This proves that is bounded. Q.E.D. (en)
dbp:title
  • Proof (en)
  • Bounded operator (en)
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dct:subject
rdfs:label
  • Bounded operator (en)
  • Beschränkter Operator (de)
  • Operador lineal acotado (es)
  • Opérateur borné (fr)
  • 有界作用素 (ja)
  • Operatore limitato (it)
  • 유계 작용소 (ko)
  • Operator liniowy ograniczony (pl)
  • Begrensde operator (nl)
  • Ограниченный оператор (ru)
  • Operador linear limitado (pt)
  • Обмежений оператор (uk)
  • 有界算子 (zh)
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