For a compact metric space (K,\rho) , the predual of Lip(K,\rho) can be identified with the normed space M(K) of finite (signed) Borel measures on K equipped with the Kantorovich–Rubinstein norm, this is due to Kantorovich [20]. Here we deduce atomic decomposition of \mathcal M(K) by mean of some results from [10]. It is also known, under suitable assumption, that there is a natural isometric isomorphism between Lip(K,\rho) and (lip(K, \rho))^{**} [15]. In this work we also show that the pair (lip(K,\rho),Lip(K,\rho)) can be framed in the theory of o–O type structures introduced by K. M. Perfekt.

We study non reflexive Orlicz spaces L Ψ and their Morse subspace M Ψ , i.e. the closure of L ∞ in M Ψ to determine when (M Ψ , L Ψ ) can be described as having an o-O type structure with respect to an equivalent norm on L Ψ . Examples of classes of Young functions for which the answer is affirmative are provided, but also examples are given to show that this is not possible for all non-reflexive Orlicz spaces. An equivalent expression of the distance in L Ψ to M Ψ , induced by the new norm, is also provided.

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Recently there has been interest in pairs of Banach spaces (E 0 , E) in an o-O relation and with E * * 0 = E.It is known that this can be done for Lipschitz spaces on suitable metric spaces. In this paper we consider the case of a compact subset K of R n with the Euclidean metric, which does not give an o-O structure, but we use part of the theory concerning these pairs to find an atomic decomposition of the predual of Lip(K). In particular, since the space M(K) of finite signed measures on K, when endowed with the Kantorovich-Rubinstein norm, has as dual space Lip(K), we can give an atomic decomposition for this space.

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