JID:FSS AID:6715 /FLA [m3SC+; v1.200; Prn:6/01/2015; 15:21] P.1(1-16) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss Fuzzy pseudo-norms and fuzzy F-spaces Sorin N˘ ad˘ aban Aurel Vlaicu University of Arad, Department of Mathematics and Computer Science, Str. Elena Dr˘ agoi 2, RO-310330 Arad, Romania Received 25 June 2013; received in revised form 10 December 2014; accepted 20 December 2014 Abstract In the present paper we firstly introduce the notion of fuzzy pseudo-norm, then we extend, improve and complete the results obtained by T. Bag and S.K. Samanta for fuzzy norms, in the fuzzy pseudo-norms context. Lastly, we introduce and discuss the notions of fuzzy F-norm and fuzzy F-space. By means of several auxiliary results, we obtain a characterization of metrizable topological linear spaces in terms of fuzzy F-norm.  2014 Elsevier B.V. All rights reserved. Keywords: Fuzzy pseudo-norm; Fuzzy F-norm; Fuzzy metric space; Fuzzy metrizable topological linear space 1. Introduction The models we work with, mathematical in their nature, must arrange themselves into pre-existing structures. In functional analysis, the fundamental structure is that of a topological linear space, but its degree of generality is much too high. Consequently, many of the important results in functional analysis have been obtained on Banach spaces (complete normed linear spaces). A significant number of familiar and useful topological linear spaces have a natural metric structure and are complete. Nevertheless, this metric does not come from a norm. These are Fréchet spaces, a term introduced by S. Banach in honour of M. Fréchet. Today, the term Fréchet space is used for a particular class of metrizable topological linear spaces, namely for the locally convex ones, while the term F- space is used for complete metrizable topological linear spaces. The topology of an F-space can be given by means of an F-norm. The foundations of fuzzy functional analysis were laid by A.K. Katsaras, who studied fuzzy topological linear spaces in his works [10,11]. Moreover, A.K. Katsaras was the first to introduce the notion of fuzzy norm – of a Minkowski type – on a linear space, associated to an absolutely convex (convex and balanced) absorbing fuzzy set. From A.K. Katsaras onward, many mathematicians have proposed several notions of fuzzy norm from different points of view. Thus, in 1992, C. Felbin [5] introduced the idea of fuzzy norm on a linear space by assigning a fuzzy real number to each element of the linear space. In 2003, following S.C. Cheng and J.N. Mordeson [4], T. Bag and S.K. Samanta [2] proposed another concept of fuzzy norm. The Bag–Samanta fuzzy norm type has proved to be the most adequate of all, even though it can be still polished, simplified, improved or generalized (see [1,7,14]). We must E-mail address: snadaban@gmail.com. http://dx.doi.org/10.1016/j.fss.2014.12.010 0165-0114/ 2014 Elsevier B.V. All rights reserved.