Difference between revisions of "Quasi-normed space"
From Encyclopedia of Mathematics
(Texify) |
(add tex done template) |
||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
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 | 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 | ||
Revision as of 06:01, 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