Tangential question that's been nagging me; in spaces of functions defined pointwise, is there a natural way to lift algebraic structures (group/ring/whatever) from the base set up to the function space? It's generally true across many separate objects in math, but I don't know a clear explanation of why. (Category theory, I assume).
Many of the propositions in the author's Appendix A are of this form.
I.e., if you look at how addition on function spaces is defined pointwise, (f+g)(x) = f(x)+g(x) -- that's different meanings of (+) on either side -- that looks exactly like the defining relation of a group homomorphism, except that the symbols are backwards.
Many of the propositions in the author's Appendix A are of this form.
I.e., if you look at how addition on function spaces is defined pointwise, (f+g)(x) = f(x)+g(x) -- that's different meanings of (+) on either side -- that looks exactly like the defining relation of a group homomorphism, except that the symbols are backwards.