Professor Themistocles M. Rassias’ special predilection and contribution to the study of mӧbius transformations is well known. möbius transformations of the open unit disc of the complex plane and, more generally, of the open unit ball of any real inner product space, give rise to möbius addition in the ball. The latter, in turn, gives rise to möbius GYRO VECTOR SPACES that enable the Poincaré ball model of hyperbolic geometry to be approached by GYRO VECTOR SPACES, in full analogy with the common VECTOR space approach to the standard model of Euclidean geometry. The purpose of this paper, dedicated to Professor Themistocles M. Rassias, is to employ the möbius GYRO VECTOR SPACES for the introduction of the hyperbolic square in the Poincare ball model of hyperbolic geometry. We will find that the hyperbolic square is richer in structure than its Euclidean counterpart.

In this paper, we have defined and studied a generalized form of topological VECTOR SPACES called s-topological VECTOR SPACES. s-topological VECTOR SPACES are defined by using semi-open sets and semi-continuity in the sense of Levine. Along with other results, it is proved that every s-topological VECTOR space is generalized homogeneous space. Every open subspace of an s-topological VECTOR space is an s-topological VECTOR space. A homomorphism between s-topological VECTOR SPACES is semi-continuous if it is s-continuous at the identity.

In this corrigenda, we have pointed out that Example 2.7, Corollary 3.7 and Corollary 5.3 in the paper: $\omega b-$Topological VECTOR SPACES, WSEAS Trans. Math. 19 (2020), $119-132$, by Latif are incorrect. We have also presented the corrected version of these results. Furthermore, we introduce and study some new classes of topological VECTOR SPACES.

n this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α , β )-metric admits no concurrent VECTOR fields. We also prove that an L-reducible Finsler metric admitting a concurrent VECTOR field reduces to a Landsberg metric. In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α , β )-metric admits no concurrent VECTOR fields. We also prove that an L-reducible Finsler metric admitting a concurrent VECTOR field reduces to a Landsberg metric. In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α , β )-metric admits no concurrent VECTOR fields. We also prove that an L-reducible Finsler metric admitting a concurrent VECTOR field reduces to a Landsberg metric. In this paper, we prove that a non-Riemannian isotropic Berwald metric or a non-Riemannian (α , β )-metric admits no concurrent VECTOR fields. We also prove that an L-reducible Finsler metric admitting a concurrent VECTOR field reduces to a Landsberg metric.

In this paper we introduce a new extension to Hahn-Banach Theorem and consider its relation with the linear operatres. At the end we give some applications of this theorem.

Among the $2$-plectic structures on VECTOR SPACES, the canonical ones and the $2$-plectic structures induced by the Killing form on semisimple Lie algebras are more interesting. In this note, we show that the group of linear preservers of the canonical $2$-plectic structure is noncompact and disconnected and its dimension is computed. Also, we show that the group of automorphisms of a compact semisimple Lie algebra preserving its $2$-plectic structure, is compact. Furthermore, it is shown that the $2$-plectic structure on a semisimple Lie algebra $\mathfrak{g}$  is canonical if and only if it has an abelian Lie subalgebra whose dimension satisfies in a special condition. As a consequence, we conclude that the $2$-plectic structures induced by the Killing form on some important classical Lie algebras are not canonical.

Let T be a bounded operator on the Banach space X and j be an analytic self-map of the unit disk D. We investigate some operator theoretic properties of bilateral composition operator Cj, T: f®T o f o j on the VECTOR-valued Hardy space Hp (X) for 1£p£+¥. Compactness and weak compactness of Cj, T on Hp (X) are characterized and when p = 2, a concrete formula for its adjoint is given.

The notion of generalized locally bounded I-topological VECTOR SPACES is introduced. Some of their important properties are studied. The relationship between this kind of SPACES and the locally bounded I-topological VECTOR SPACES introduced by Wu and Fang [Boundedness and locally bounded fuzzy topological VECTOR SPACES, Fuzzy Math. 5 (4) (1985) 87-94] is discussed. Moreover, we also use the family of generalized fuzzy quasi-norms to characterize the generalized locally bounded I-topological VECTOR SPACES, and give some applications of this characterization.