Abstract. Bretagnolle et al. [4] showed that for $1\leq p\leq2$, a normed real vector space $(E,\|\cdot\|)$ can be embedded in an $L^p$-space if and only if $\|\cdot\|^p$ is negative definite. Recently, the author proved that the requirement that $\|\cdot\|$ be a norm can be replaced with the conditions that $\|tx\|=|t| \|x\|$ for $t\in{\msbm R}$ and $x\in E$ and $\|x\|>0$ for $x\in E\setminus\{0\} $, and that the result so modified holds whenever $0< p< 2$. Zoltán Sasvári then showed that for any $p>0$ which is not an even integer, $(E,\|\cdot\|)$ can be embedded in an $L^p$-space if and only if $\|\cdot\|$ is continuous on finite-dimensional linear subspaces of $E$ and $(-1)^k\|\cdot\|^p\in P(1,k)$ (see definitions below) where $k=\lceil p/2\rceil $. Finally, the author found a necessary and sufficient condition for the case $p=2k$, and proved that it is not necessary to assume continuity on finite-dimensional linear subspaces.

AMS Subject Classification (1991): 28-XX, 43A35, 46-XX, 51-XX

Received July 15, 1996. (Registered under 6108/2009.)