. FA ] 5 O ct 2 00 5 A DIRECT SUM DECOMPOSITION FOR DUAL SPACES

We developpe a direct sum decomposition for n-dual spaces. Introduction Let X be a Banach space. We will denote π0 : X → X ∗∗ and π1 : X ∗ → X∗∗∗ the canonical maps ofX and its dual space X∗ into their respective bidual spaces. According to [G] we have X∗∗∗ = Ran(π1)⊕Ker(π ∗ 0), where π∗ 0 denotes the adjoint map of π0. 1. Remark. In [G] we also saw as an application that X is reflexive if and only if its dual space X∗ is reflexive. And since the dual space of the latter is the bidual space of X , in thiis case we conclude that X∗∗ is reflexive also. And so on, we may induce that X is reflexive if and only if for all n the nth-dual space is reflexive. As the reader can see the notation becomes cumbersome when we consider the nth-dual space. X∗···∗ with ∗ · · · ∗ n-times And since the rest of the paper deals with these, we will use the following notation. Let N denote the natural numbers with 0 included. 2. Definition. Let X be a Banach space. For n ∈ N, n ≥ 1 denote X the nth−dual space of X . Then we will denote its canonical map into its bidual space πn : X ∗n → X with its adjoint map π∗ n : X → X Note that these dual spaces are all Banach spaces, and that we have the maps πn : X ∗n → X π∗ n−1 : X ∗(n+2) → X and since the cited result is valid for them we state the result for sake of completeness as 3. Proposition. For n ∈ N, n ≥ 1. X = Ran(πn)⊕Ker(π ∗ n−1) 1991 Mathematics Subject Classification. Primary 46B10.