The L-FUZZY APPROXIMATION operator associated with an L-FUZZY APPROXIMATION space (X, R) turns out to be a saturated L-FUZZY closure (interior) operator on a set X precisely when the relation R is reflexive and transitive. We investigate the relations between L-FUZZY APPROXIMATION SPACES and L-(fuzzy) topological SPACES.

The main purpose of this paper is to consider the t-best simultaneous APPROXIMATION in fuzzy normed SPACES. We develop the theory of t-best simultaneous APPROXIMATION in quotient SPACES. Then, we discuss the relationship in t-proximinality and t-Chebyshevity of a given space and its quotient space.

The main purpose of this paper is to find t-best APPROXIMATIONs in fuzzy normed SPACES. We introduce the notions of t-proximinal sets and F-APPROXIMATIONs and prove some interesting theorems. In particular, we investigate the set of all t-best APPROXIMATIONs to an element from a set.

APPROXIMATION SPACES, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by Butzer and Scherer in 1968 and, independently, by Y. Brudnyi and N. Kruglyak in 1978, and popularized by Pietsch in his seminal paper of 1981. Pietsch was interested in the parallelism that exists between the theories of APPROXIMATION SPACES and interpolation SPACES, so that he proved embedding, reiteration and representation results for APPROXIMATION SPACES. In particular, embedding results are a natural part of the theory since its inception. The main goal of this paper is to prove that, for certain classes of APPROXIMATION schemes (X, {An}) and sequence SPACES S, if S1ÌS2Ìc0 (with strict inclusions) then the APPROXIMATION space A(X, S1, {An}) is properly contained into A(X, S2, {An}). We also initiate a study of strict inclusions between interpolation SPACES, for Petree’s real interpolation method.

The purpose of this paper is to introduce and to discuss the concept of p-APPROXIMATION and p-orthogonality in vector SPACES, and to obtain some results on p- orthogonality in vector SPACES similar to some well known results on the orthogonality in normed SPACES. We also discuss the concept of p-extension of linear functionals on a vector space, and give a characterization of linear functionals on a subspace having a unique p-extension Hahn-Banach to the whole vector space.

We study relations between evidence theory and S-APPROXIMATION SPACES. Both theories have their roots in the analysis of Dempster’ s multivalued mappings and lower and upper probabilities, and have close relations to rough sets. We show that an S-APPROXIMATION space, satisfying a monotonicity condition, can induce a natural belief structure which is a fundamental block in evidence theory. We also demonstrate that one can induce a natural belief structure on one set, given a belief structure on another set, if the two sets are related by a partial monotone S-APPROXIMATION space.

Let X be a Banach space and let Y be a separable Lindenstrauss space. We describe the Banach space P (Y, X) of absolutely summing operators as a general L1-tree space. We also characterize the bounded APPROXIMATION property and its weak version for X in terms of the space of integral operators I (X, Z*) and the space of nuclear operators N (X, Z*), respectively, where Z is a Lindenstrauss space, whose dual Z* fails to have the Radon-Nikod´ym property.