In this paper, we have generalized the definition of the vector SPACE by considering the group as a canonical m-ary hypergroup, the field as a krasner (m; n)-hyperfield and considering the multiplication structure of a vector by a scalar as hyperstructure. Also we will be consider a normed m-ary HYPERVECTOR SPACE and introduce the concept of convergent of sequence on m-ary hypernormed SPACEs and bundle subset.

The aim of this paper is to prove the Uniform Boundedness Principle and Banach-Steinhaus Theorem for anti linear operators and hence strong linear operators on Banach HYPERVECTOR SPACEs. Also we prove the continuity of the product operation in such SPACEs.

The main purpose of this paper is to study normed HYPERVECTOR SPACEs. We generalize some definitions such as basis, convexity, operator norm, closed set, Cauchy sequences, and continuity in such SPACEs and prove some theorems about them.

In this paper we investigate the algebraic properties of dimension of fuzzy HYPERVECTOR SPACEs. Also, we prove that two isomorphic fuzzy HYPERVECTOR SPACEs have the same dimension.

The purpose of this paper is the study of sub-hyperSPACEs of a HYPERVECTOR SPACE investigating some of their properties. In this regard, the notion of a normal fuzzy sub-hyperSPACE of HYPERVECTOR SPACE is introduced. Using it, we construct new fuzzy sub-hyperSPACEs. We also show that, under certain conditions, a fuzzy sub-hyperSPACE of a HYPERVECTOR SPACE is two-valued and takes the values 0 and 1.

In this paper, two related quotient structures are investigated utilizing the concept of coset. At first, a new HYPERVECTOR SPACE F/V = (F/V,\circ,\circledcirc,K) is created, which is composed of all cosets of a bipolar fuzzy soft set (F;A) over a HYPERVECTOR SPACE V . Then it will be shown that dim F/V = dim V/W, where the quotient HYPERVECTOR SPACE V/W includes all cosets of an especial subhyperSPACE W of V. Also, three bipolar fuzzy soft sets over the quotient HYPERVECTOR SPACE V/W are presented and in this way some new bipolar fuzzy soft HYPERVECTOR SPACEs are defined.

The aim of this paper is the study of fuzzy basis and dimension of fuzzy HYPERVECTOR SPACEs. In this regard, first the notions of fuzzy linear independence and fuzzy basis are introduced and then some related results are obtained. In particular, it is shown that for a large class of fuzzy HYPERVECTOR SPACE the fuzzy basis exist. Finally, dimension of a fuzzy HYPERVECTOR SPACE is defined and the basic properties of that are investigated.

In this note first we shall redefine the notion of a fuzzy HYPERVECTOR SPACE (see [1]) and then we introduce some further concepts of fuzzy HYPERVECTOR SPACEs, such as convex fuzzy sets and balance fuzzy sets in fuzzy HYPERVECTOR SPACEs over valued fields. Finally, we briefly discuss on convex (balanced) hull of a given fuzzy set of a HYPERVECTOR SPACE.

The purpose of this paper is to introduce and study hyperframes. Where a frame is a generlization of a basis of a vector SPACE, a hyperframe will act as a generalization of a basis in HYPERVECTOR SPACE. The present research will only considerl HYPERVECTOR SPACEs over the reals, viewed as a Krasner hyperfield. In particular, similarity, equivalency, and dual hyperframes are discussed.