On conjugate spaces of Nakano spaces

Abstract

The Nakano space is a normed space. The relations between the modular and its norms are the most important problem in the theory of Nakano spaces. To determine the types of modulars whose norms are proportional has been completely solved by Amemiya [1 ] and Yamamuro [1]. The purpose of this paper is to consider the relations from another point of view, namely, as the relations between Young's inequality and H6lder's inequality. It is well known that the fundamental inequality in the theory of normed linear spaces is that of H6lder, and, if we consider the conjugate spaces of them, it is important to know when the Holder's inequality takes equality sign. On the other hand, in the theory of Nakano spaces, the fundamental inequality is that of Young. Therefore, it is natural to study the relations between values at which the inequalities take equality signs. This paper consists of three sections and two appendices. In ?1, as preparation to the following sections, we will restate without proofs some results of Nakano [I]. In ?2, we will show the equivalence of the reflexivity of the modular and that of its norms. Concerning the reflexivity of norms, Mori-Amemiya-Nakano's theorem [1 ] is fundamental and its proof is simple. At first, by using this theorem, we will give a new and simple proof of Nakano's reflexivity theorem of modular. Next, we will write a method by which we can prove the reflexivity of norms by that of modular. This method has been known among the group conducted by Professor Nakano, but has not yet been published. In ?3, we will calculate values of norms of a linear functional which plays important roles in the theory of Nakano spaces. In connection with the linear functional, we will give more information in Appendix 1. In Appendix 2, we will consider strict convexity of modulars and their norms.