This is a continuation of ibid. 6, 85-95 (1981;

Zbl 0463.46009). It is shown that a topology

$\tau $, on a vector space E, is linear iff the fuzzy topology

$\omega $ (

$\tau )$, consisting of all

$\tau $-lower semicontinuous fuzzy sets, is linear. The fuzzy seminormed and the fuzzy normed linear spaces are introduced and some of their properties are studied. It is also given the concept of a bounded fuzzy set and the concept of a bornological fuzzy linear space. A linear topology

$\tau $ on E is bornological iff

$\omega $ (

$\tau )$ is bornological. The locally convex fuzzy linear topologies form a special class of fuzzy linear topologies. Some of the properties of these spaces as well as of the quotient spaces and the bornological spaces, are investigated.