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Orders of Infinity

Many problems in analysis (as well as adjacent fields such as combinatorics, theoretical computer science, and PDE) are interested in the order of growth (or decay) of some quantity $latex {X}&…

This can be compared with the situation with, say, the Fréchet filter on the natural numbers , in which one has to insert some qualifier such as “after passing to a subsequence if necessary” in order to make the above axiom true. With these operations, combined with the ultrafilter axiom, we see that obeys the laws of many standard algebraic structures, the proofs of which we leave as exercises for the reader: Ordered log-vector spaces may seem like a strange and exotic concept, but they are actually already studied implicitly in analysis, albeit under the guise of other names such as log-convexity.

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