Turkish Journal of Mathematics




In this paper we consider a particular type of modular sequence spaces defined with the help of a given sequence \alpha = {\alpha_n} of strictly positive real numbers \alpha_n's and an Orlicz function M. Indeed, if we define M_n(x) = M(\alpha_nx) and \tildeM_n(x)=M(x/\alpha_n), x\in[0, \infty), we consider the modular sequence spaces l{M_n} and l{\tildeM_n}, denoted by l_M^{\alpha} and l_{\alpha}^M respectively. These are known to be BK-spaces and if M satisfies \Delta_2-condition, they are AK-spaces as well. However, if we consider the spaces l_{\alpha}^M and l_n^{\alpha} corresponding to two complementary Orlicz functions M and N satisfying \Delta_2-condition, they are perfect sequence spaces, each being the Köthe dual of the other. We show that these are subspaces of the normal sequence spaces \mu and \eta which contain \alpha and \alpha^{-1}, respectively. We also consider the interrelationship of l_{\alpha}^M and l_M^{\alpha} for different choices of \alpha.

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