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Difference between revisions of "Quasi-normed space"

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A [[Linear space|linear space]] on which a [[Quasi-norm|quasi-norm]] is given. An example of a quasi-normed space that is not normed is the [[Lebesgue space|Lebesgue space]] $L_p(E)$ with $0<p<1$, in which a quasi-norm is defined by the expression
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A [[Linear space|linear space]] on which a [[Quasi-norm|quasi-norm]] is given. An example of a quasi-normed space that is not normed is the [[Lp spaces|Lebesgue space]] $L_p(E)$ with $0<p<1$, in which a quasi-norm is defined by the expression
  
 
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Latest revision as of 21:36, 29 May 2016

A linear space on which a quasi-norm is given. An example of a quasi-normed space that is not normed is the Lebesgue space $L_p(E)$ with $0<p<1$, in which a quasi-norm is defined by the expression

$$ \| f \|_p = \left[ \int_E |f(x)|^p \; dx \right]^{1/p}, \quad f \in L_p(E) $$


Comments

The quasi-normed topological vector spaces are precisely the locally bounded topological vector spaces, cf. Quasi-norm.

How to Cite This Entry:
Quasi-normed space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-normed_space&oldid=38874
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article