Abstract. This paper is the outcome of a workshop held in Rome in November 2011 on the occasion of the 25th anniversary of the POEM (Physical Oceanography of the Eastern Mediterranean) program. In the workshop discussions, a number of unresolved issues were identified for the physical and biogeochemical properties of the Mediterranean Sea as a whole, i.e., comprising the Western and Eastern sub-basins. Over the successive two years, the related ideas were discussed among the group of scientists who participated in the workshop and who have contributed to the writing of this paper.Three major topics were identified, each of them being the object of a section divided into a number of different subsections, each addressing a specific physical, chemical or biological issue:1. Assessment of basin-wide physical/biochemical properties, of their variability and interactions.2. Relative importance of external forcing functions (wind stress, heat/moisture fluxes, forcing through straits) vs. internal variability.3. Shelf/deep sea interactions and exchanges of physical/biogeochemical properties and how they affect the sub-basin circulation and property distribution.Furthermore, a number of unresolved scientific/methodological issues were also identified and are reported in each sub-section after a short discussion of the present knowledge. They represent the collegial consensus of the scientists contributing to the paper. Naturally, the unresolved issues presented here constitute the choice of the authors and therefore they may not be exhaustive and/or complete. The overall goal is to stimulate a broader interdisciplinary discussion among the scientists of the Mediterranean oceanographic community, leading to enhanced collaborative efforts and exciting future discoveries.

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O. IntroductionIf E is a nuclear space, Grothendieck [1] proved that for every 0-neighborhood U in E, there is an absolutely convex 0-neighborhood V in E, Vc U such that •v is norm-isomorphic to a subspace of l ~ (whatever pE[1, oo)). As usual, Ev is the completion of the linear space I~)=E/A n-iV normed with the gauge of V.n If E is the nuclear space s of rapidly decreasing sequences and F is any infinitedimensional Banach space with Schauder basis, then for every 0-neighborhood U in E there exists an absolutely convex 0-neighborhood Vc U in E such that /~v is norm-isomorphic to F (see [6]). This result of Saxon was improved later on by Valdivia [8] proving its validity when E is an arbitrary nuclear space and F any infinite-dimensional separable Banach space.In the present paper we bring up these results into the context of 2-nuclearity. Namely, we will prove that for certain sequence spaces 4, a Mackey space a can be found satisfying the following condition: If F is any infinite-dimensional Banach space with Schauder basis, for every 0-neighborhood U in o-there is an absolutely convex 0-neighborhood V in o-such that ffv is norm-isomorphic to F. As a consequence we prove an embedding theorem of k-nuclear spaces into some product of any given infinite-dimensional Banach space with Schauder basis. This research, supported by the University of Extremadura, was carried out during the author's visit to the University of Kaiserslautern (F.R.G.). Definitions and previous resultsThe linear sequence spaces 2 we will deal to are assumed throughout to be normal, additive, decreasing rearrangement invariant and such that 2ci q for some q>0 (see [4] for definitions). The diametral dimension of a Hausdorff locally convex space E, denoted by A (E), is the collection of all sequences (6n) of nonnegative numbers such that for every 0-neighborhood U in E there is a 0-neighborhood V in E, VcU such that On(V, U)<=M6n for every nEN and for some M_->0. Here 6n(V, U) is the n-th Kolmogoroff diameter of V with respect to U. This notion of diametral dimension, different of that of Terzioglu [7], fits better in our context of 2-nuclearity which is defined as follows: A Hausdorff locally convex space E is said to be 2-nuclear if for each p>0 the following condition holds: For every 0-neighborhood U in E there is a 0-neighborhood V in E, Vc U such that (~,(V, U))E2P (a sequence (~n) is in 2p iff (~) is in 4).Following [4] we denote by 4 + the subset of all sequences (~,) in 2 such that 1"

A HAUSDORFF locally convex space is said to be c,-barrelled (respectively cu-barrelled) if each sequence in the dual space t h a t converges weakly to 0 (res!,r:ctively t h a t is weakly ?.~oundecl), is equicontinuous. It is proved that if a c,,-barrelled space E has dual E' weakly sequentially complete, then every subsi'ace of countable codimensjon of E is c,-barrcllecl. %Ire prove that the hypothesis on E' cannot be dropped and we supply a n esniiiple of a complete c,,-hnrrellecl space with dual wealily sequentially coin1)lete that is not co-barrelled. Introdnetion and PreliriiinnriesFor the notations and terminology as well as for the not proved results, we will refer throughout t o [ l], [4] a i d [3].A HAVSDORFP locally convex space [E, t] or simply E with topological ctual E' is said t o be c,-barrelled (resp. co-barrelled) if each sequence t h a t converges t o 0 (resp. t h a t is bounded) in [E', a@', E ) ] is equicontinuous. E has the property (S) if [E', a@', E ) ] is sequentially complete. It is known [&] t h a t every subspace of countable coclimension of a w-barrelled space is w-barrelled. The proof of this inheritance property relies heavily on the sequential coinpleteness of [E', o(E', E ) ] . However, if E is c,,-barrelled, E does not posses necessarily the property (8) and E might contain (and in fact does) countable-codimensional subspaces that are not c,,-barrelled. I n the main result we prove that every countable-codimensional subspzce of a co-barrelled space with property (8) is c,-barrelled and we furnish an example showing t h a t the property (S) cannot be dropped.As usual, we will denote by p(E, E') the MACKEY topology associated to the chal pair {E, B'). We denote by to(&', E') the locally convex topology on E of the zcniform convergence o n all the a(E', E)-convergent to 0 sequences of E'. Obviously a locally convex space [ E , z] with dual E is c,-barrelled iff to(E, E') SZ. Moreover, for the ;ILACKEY topology we have 1) The author was supported by a Fulbright-MEC fellowship. The hospitality of the Department of Mathematics, University of RIaryland, is gratefully acknowledged and the author thanks Dr. J. H O R V~T H for his helpful comments. Current address: Departamento Matemiticas, Facultad cle Ciencias. 06071-Badajoz, Spain.