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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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076620/q0766201.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076620/q0766202.png" />, 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 [[Lebesgue space|Lebesgue space]] $L_p(E)$ with $0<p<1$, in which a quasi-norm is defined by the expression
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076620/q0766203.png" /></td> </tr></table>
 
  
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$$
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\| f \|_p = \left[ \int_E |f(x)|^p \; dx \right]^{1/p}, \quad f \in L_p(E)
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$$
  
  
 
====Comments====
 
====Comments====
 
The quasi-normed topological vector spaces are precisely the locally bounded topological vector spaces, cf. [[Quasi-norm|Quasi-norm]].
 
The quasi-normed topological vector spaces are precisely the locally bounded topological vector spaces, cf. [[Quasi-norm|Quasi-norm]].

Revision as of 06:00, 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=17613
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article