Some properties of equivalent fuzzy norms

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 <TEX>$\rho_1,\;\rho_2$</TEX> are equivalent fuzzy norms on a linear space, then for every <TEX>$\varepsilon\in(0.1)$</TEX>, the induced crisp norms <TEX>$P_\varepsilon^1,\;and\;P_\varepsilon^2$</TEX>, 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 <TEX>$\rho$</TEX> be a lower semicontinuous fuzzy norm on a normed linear space X, and have the bounded support. Then <TEX>$\rho$</TEX> is equivalent to the fuzzy norm <TEX>$\chi_B$</TEX> where B is the closed unit ball of X. The lower semi-continuity of <TEX>$\rho$</TEX> is an essential condition which guarantees the continuity of <TEX>$P_\varepsilon$</TEX>, where 0 < e < 1. As the last result, we prove that : if <TEX>$\rho$</TEX> is a fuzzy norm on a finite dimensional vector space, then <TEX>$\rho$</TEX> is equivalent to <TEX>$\chi_B$</TEX> if and only if the support of <TEX>$\rho$</TEX> is bounded.