# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Fuzzy topological vector spaces. II. (English) Zbl 0555.46006
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.

##### MSC:
 46A99 Topological linear spaces 54A40 Fuzzy topology 46A08 Barrelled spaces, bornological spaces 46A03 General theory of locally convex spaces
Full Text:
##### References:
 [1] Chang, C. L.: Fuzzy topological spaces. J. math. Anal. appl 24, 182-190 (1968) · Zbl 0167.51001 [2] Katsaras, A. K.; Liu, D. B.: Fuzzy vector spaces and fuzzy topological vector spaces. J. math. Anal. appl 58, 135-146 (1977) · Zbl 0358.46011 [3] Katsaras, A. K.: Fuzzy topological vector spaces I. Fuzzy sets and systems 6, 85-95 (1981) · Zbl 0463.46009 [4] Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. math. Anal. appl 56, 621-633 (1976) · Zbl 0342.54003 [5] Lowen, R.: Convex fuzzy sets. Fuzzy sets and systems 3, 291-310 (1980) · Zbl 0439.52001 [6] Pao-Ming, Pu; Ying-Ming, Liu: Fuzzy topology I. Neighborhood structure of a point and Moore-Smith convergence. J. math. Anal. appl 76, 571-599 (1980) · Zbl 0447.54006 [7] Warren, R. H.: Neighborhoods bases and continuity in fuzzy topological spaces. Rocky mountain J. Math 8, 459-470 (1978) · Zbl 0394.54003 [8] Warren, R. H.: Fuzzy topologies characterized by neighborhood systems. Rocky mountain J. Math 9, 761-764 (1979) · Zbl 0429.54003 [9] Wong, C. K.: Fuzzy topology: product and quotient theorems. J. math. Anal. appl 45, 512-521 (1974) · Zbl 0273.54002 [10] Zadeh, L. A.: Fuzzy sets. Information and control 8, 338-353 (1965) · Zbl 0139.24606