We modify the very well known theory of normed spaces (E, · ) within functional analysis by considering a sequence ( · n : n ∈ N) of norms, where · n is defined on the product space E n for each n ∈ N.Our theory is analogous to, but distinct from, an existing theory of 'operator spaces'; it is designed to relate to general spaces L p for p ∈ [1, ∞], and in particular to L 1 -spaces, rather than to L 2 -spaces.After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory, that we shall use, we shall present in Chapter 2 our axiomatic definition of a 'multi-normed space' ((E n , · n ) : n ∈ N), where (E, · ) is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum, maximum, and (p, q)-multi-norms based on a given space. Multi-norms measure 'geometrical features' of normed spaces, in particular by considering their 'rate of growth'. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators.A substantial number of examples of multi-norms will be presented.Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to 'multi-topological linear spaces' through 'multi-null sequences', and to 'multi-bounded' linear operators, which are exactly the 'multi-continuous' operators. We define a new Banach space M(E, F ) of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices.We conclude with a theory of 'orthogonal decompositions' of a normed space with respect to a multi-norm, and apply this to construct a 'multi-dual' space.Applications of this theory will be presented elsewhere.

Carbon dioxide can be used in various ways as a cheap C1 source. However, the utilization of CO2 requires energy or energy-rich reagents, which leads to further emissions, and therefore, diminishes the CO2-saving potential. Therefore, life cycle assessment (LCA) is required for each process that uses CO2 to provide valid data for CO2 savings. Carbon dioxide can be incorporated into epoxidized fatty acid esters to provide the corresponding carbonates. A robust catalytic process was developed based on simple halide salts in combination with a phase-transfer catalyst. The CO2-saving potential was determined by comparing the carbonates as a plasticizer with an established phthalate-based plasticizer. Although CO2 savings of up to 80 % were achieved, most of the savings arose from indirect effects and not from CO2 utilization. Furthermore, other categories have been analyzed in the LCA. The use of biobased material has a variety of impacts on categories such as eutrophication and marine toxicity. Therefore, the benefits of biobased materials have to be evaluated carefully for each case. Finally, interesting properties as plasticizers were obtained with the carbonates. The volatility and water extraction could be improved relative to the epoxidized system.

Let G be a locally compact group, and let L1 (G) be the Banach algebra which is the group algebra of G. We consider a variety of Banach left L1 (G)‐modules over L1 (G), and seek to determine conditions on G that determine when these modules are either projective or injective or flat in the category. The answers typically involve G being compact or discrete or amenable. For example, in the case where G is discrete and 1 < p < ∞, we find that the module ℓp (G) is injective whenever G is amenable, and that, if it is amenable, then G is ‘pseudo‐amenable’, a property very close to that of amenability. 2000 Mathematics Subject Classification 46H25, 43A20.

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