Recent content by lugita15

“I am not a number theorist, and the author of this post is clearly more knowledgable than I am.” Actually I know nothing at all about elliptic curves. It’s just that this question arose in my research as a logician.

##\mathbb{Q}(\zeta_{p^\infty})##, also written as ##\mathbb{Q}(\mu_{p^\infty})## or ##\mathbb{Q}(p^\infty)##, denotes ##\mathbb{Q}## adjoined with the ##p^{n}##th roots of unity for all ##n##. It's the union of a cylotomic tower, and it's studied in subjects like Iwosawa theory and class field...

Hi Demystifier, nice to talk to you after ten years. Anyway, what would you say about counterfactual definiteness?

It’s a standard notion in functional analysis. If ##X## is a topological vector space, ##\{x_i: i\in I\}\subseteq X##, and ##x\in X##, then we say that the unordered sum ##\Sigma_{i\in I}x_i## converges to ##x## if for every open set ##U## containing ##x##, there exists a finite set...

Sorry, there’s a typo in my question, it should say that f is a function from Int(P) to F. And yeah, after I posted my question I realized that if A is an uncountable set and F is a topological field, then the set X of functions from A to F is a topological vector space with the topology of...

Yeah, that’s why I put “uncountable Schauder basis” in quotes. Because the property that I(P) has is akin to a Schauder basis except it involves uncountable linear combinations rather than countable linear combinations.

Let ##P## be an uncountable locally finite poset, let ##F## be a field, and let ##Int(P)=\{[a,b]:a,b\in P, a\leq b\}##. Then the incidence algebra $I(P)$ is the set of all functions ##f:P\rightarrow F##, and it's a topological vector space over ##F## (a topological algebra in fact) with an...

I’m not interested in normed vector spaces at all. As Wikipedia says “Schauder bases can also be defined analogously in a general topological vector space.”

We usually talk about ##F[[x]]##, the set of formal power series with coefficients in ##F##, as a topological ring. But we can also view it as a topological vector space over ##F## where ##F## is endowed with the discrete topology. And viewed in this way, ##\{x^n:n\in\mathbb{N}\}## is a...

Thanks, I fixed it.

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...

Thanks, that makes sense.

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...

Yes, if we define a partial order on the set of all functions infinitely differentiable at 0 by saying that f<g if the limit of f(x)/g(x) as x goes to 0 = 0, and then take a maximal totally ordered subset of that partially ordered set, then the order type of that set will be much bigger than the...

Let ##X## be the set of all functions infinitely differentiable at ##0##. Let's define an equivalence relation on $X$ by saying that ##f\sim g## if there exists a sufficiently small open interval ##I## containing ##0## such that ##f(x)=g(x)## for all ##x## in ##I##. Then the set of germs of...