math and computers and stuff

One fun fact / alternative perspective that occurred to me as I was working through some elementary problems on big O notation: if viewed as a class of functions, each big-O set {f(x)|f(x)=O(g(x))} is a linear subspace of the function space under consideration. For instance, if we’re looking only at polynomials of degree <= n, as subspaces O(1) in O(x) in O(x^2) in … gives us a flag of subspaces. I suppose one could use this to draw pictures of the sets of functions that satisfy certain relations. I wonder if this would be helpful to explain the concept, or if it just makes things more confusing.

It would be great if this told us something cool for more general spaces of functions, but of course they’re all infinite-dimensional so it’s hard to say much about what’s happening, other than that I suspect it’s a drastically reduced subset. It’s hard to think of a nice map either way. (If we’re being silly about things, the set of functions that actually come up in CS theory is basically finite-dimensional anyways!)

Note: big theta is a bit difference as it’s more like equivalence relations / equivalence classes than it is like a linear subspace.

Disclaimer: I’m not really sure this is right/sensible. Usually we only talk about big O for the positive side of things, so I’m using a somewhat vague/made-up more general definition that probably just boils down to some sort of Lipschitz condition. Please forgive me as this is all speculative and I’m not an analyst by training.