If you stick with formal sums -- i.e., sequences of reals such as R^oo (direct product) -- without trying to assign a real number to each, you should be fine. Then if R^oo is indexed by Z (rather than say Z+), we have the double-ended sequences. And index-shifting can define an equivalence relation on R^oo whose quotient might be some interesting algebraic structure, replete with addition & scalar multiplication . . . I think. But if you do try to assign all elements of R^oo or even Hilbert space an extension of the sum-of-coordinates functions (from absolutely convergent double-ended series, where it is clearly well-defined), I think you always run into trouble if you want the definition to be non-random. But I'm not certain how to formulate or prove this precisely. --Dan David W. wrote: << With regard to the double-ended sums, I was hoping to treat them as a formalism, and avoid issues of convergence (which end do you converge toward?). Thus we would assign a reasonable value to some class of basic expressions (e.g, ... + 0 + 0 + 0 + 9 + 0 + 0 + 0 + ... = 9), and apply some formal arithmetic rules (value preserved by shifting, term-by-term sum, term-by-term scalar multiplication) and see where that leads us (I expect to a contradiction).

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Daniel Asimov Visiting Scholar Department of Mathematics University of California Berkeley, California