Abstract
In the present paper, we observe a relation between fuzzy norms and induced crisp norms on a linear space. We first prove that if $\rho_1,\;\rho_2$ are equivalent fuzzy norms on a linear space, then for every $\varepsilon\in(0.1)$, the induced crisp norms $P_\varepsilon^1,\;and\;P_\varepsilon^2$, respectively are equivalent. Since the converse does not hold, we prove it under some strict conditions. And consider the following theorem proved in [8]: Let $\rho$ be a lower semicontinuous fuzzy norm on a normed linear space X, and have the bounded support. Then $\rho$ is equivalent to the fuzzy norm $\chi_B$ where B is the closed unit ball of X. The lower semi-continuity of $\rho$ is an essential condition which guarantees the continuity of $P_\varepsilon$, where 0 < e < 1. As the last result, we prove that : if $\rho$ is a fuzzy norm on a finite dimensional vector space, then $\rho$ is equivalent to $\chi_B$ if and only if the support of $\rho$ is bounded.
