Difference between revisions of "Quasi-normed space"
From Encyclopedia of Mathematics
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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]] < | + | 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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| + | $$ | ||
| + | \| f \|_p = \left[ \int_E |f(x)|^p \; dx \right]^{1/p}, \quad f \in L_p(E) | ||
| + | $$ | ||
====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=38873
Quasi-normed space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-normed_space&oldid=38873
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article