In 1984, Katsaras, Wu and Fang defined two kinds of fuzzy norms on a linear space to construct the fuzzy vector topological structure on the space, independently [see C. Wu and J. Fang, “Fuzzy generalization of Kolmogoroff’s theorem”, Journal of Harbin Institute of Technology, No. 1, 1–7 (1984), in Chinese]. Later, a few mathematicians have introduced and discussed several notions of fuzzy norm from different points of view. By means of the notion of a fuzzy norm presented by A. K. Mirmostafaee and the authors of this paper, the authors also define a new notion of fuzzy inner product space, establish the fuzzy Cauchy-Schwarz inequality and prove a version of the parallelogram law in this setting.