# Quasi-normed space

From Encyclopedia of Mathematics

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://www.encyclopediaofmath.org/index.php?title=Quasi-normed_space&oldid=38878

This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article