In the present paper, we observe a relation between fuzzy norms and induced crisp norms on a linear space. We first prove that if $ ho_1,; ho_2$ are equivalent fuzzy norms on a linear space, then for every $varepsilonin(0.1)$, 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 $ ho$ be a lower semicontinuous fuzzy norm on a normed linear space X, and have the bounded support. Then $ ho$ is equivalent to the fuzzy norm $chi_B$ where B is the closed unit ball of X. The lower semi-continuity of $ ho$ is an essential condition which guarantees the continuity of $P_varepsilon$, where 0 < e < 1. As the last result, we prove that : if $ ho$ is a fuzzy norm on a finite dimensional vector space, then $ ho$ is equivalent to $chi_B$ if and only if the support of $ ho$ is bounded.