@leslietownes Just quickly coming back to this conversation -- as regards the $T$-periodic space of functions which are spanned by the trigonometric functions, I guess I gave the wrong norm right? It should be induced by an inner product looking something like $<f,g> = \lim_{T \to \infty}\frac{1}{T}\int_{-T/2}^{T/2} fg$?

EE18 if you are looking at periodic functions the usual norm of <f,g> would be an integral of fg over one period (which is arbitrary but usually fixed, e.g. [0,T] or [-T/2,T/2]). you might be able to write out some limit over increasingly large that ends up being the same thing because of the periodicity, but i don't know why you would want that

you can also define inner products of not necessarily periodic functions in terms of integrals over all of R, usually with some kind of "decay at infinity" hypothesis floating in the background to ensure that the thing makes sense and is finite, and sometimes (not always, and not usually by definition) expressions for those sorts of norms are written as limits like that

@leslietownes Ah OK I think this is what's going on, that there's some equivalence to the infinite limit version because of periodicity

I'm working with that engineering text which is keeping stuff hazy, so wanted to just clarify what was going on

@Obliv agreed, but preference should be given to consistency.

@EE18 \langle and \rangle.

@leslietownes You, too. :(

:(

@leslietownes Though in a lot of cases, those won't really be inner products, but the dual pairing (which devolves to an inner product in spaces which are self-dual. (Not that this really matters).

I personally write my inner products like this $\{\left\{\langle\left[<f,g>\right]\rangle\right\}\}$

just to be clear

more brackets means stronger seal

xander: or maybe it is the inner product that evolves into a dual pairing. hippy music

🎶🎵🎶🎸

@leslietownes That isn't how evolution works.

Though it may be the that the inner product is the distant descendant of some common ancestor of the dual pairing.

@leslietownes Actually sorry for perseverating on this Leslie, but I've been searching around and not able to find a wikipage or something like that for this. What term should I be using to learn more about this equivalence between the two possible inner products on this periodic function space?

maybe just prove it? i could see it as a homework problem in an analysis book. if f is periodic with period T then the average value of f on [-L, L] goes to the average value of f on [0,T] as L goes to infinity. it should make intuitive sense.

or look at math.stackexchange.com/questions/2768853/…

there are several duplicates of that question on the site too, all with similar looking proofs. it's maybe an expository challenge to make the argument look good. it might clean things up a little to assume T = 1 or something like that

I realized I have no idea how to glue together closed subsets bounded by analytic boundary S^1's when I'm not in the Top Cat. Do I operate in the Top Cat and then use some deformation or rigidification process to deform the topological surface into smooth/real analytic manifold? I think that is the right direction because there are theorems guranteeing a topological manifold and smooth manifold can be deformed to a anlaytic one. I'm lost on the details though

not sure where to start exactly but maybe in Top, and generate a topological surface S^2, and then sequentially layer on smooth then analytic structures. If the S^2 here is obtained through a gluing and one can obtain an analytic metric compatible with the base (unwrapped quotient), related through the section/exact projection correspondence allowing for a direct analytic quotient metric on the S^2. I think I should go for the low hanging fruit and that would be recovering a well defined

analytic manifold with a well defined analytic quotient metric