What I don't like is that somehow it feels like if I can use a normalized version of $\hat z$ I shouldn't need to keep track of the norm of $z$ at all, making the twist unnecessary---which would be wrong.
But I don't see a real problem with the actual map I wrote down...
What if I'd just say I map $([z],v)\mapsto d_{\hat z}\pi(v)$, eliminating the dependence on the norm of the representative by always picking the norm-1 one? Is it somehow not possible to keep track of the norm of the representative of $[z]$?
I don't see what you mean by "in a specific dimension", because we're fixing the dimension of the base sphere (assuming of course that it was to me and not Danu).
@MikeMiller What would you say is the significance of the following result in Holomorphic Functional Calculus: If $a \in \mathcal{A}:= \text{Banach Algebra}$ and $f \in \mathcal{Hol}(a)$. Then $$\sigma(f(a)) = f(\sigma(a)).$$
@MikeMiller I discussed operator theory with you some time ago so I assumed you might know. We are being presented with Banach Algebra definitions and theorems and then Holomorphic Functional Calculus and then Continuous Function Calculus...We have not covered Continuous Function Calculus yet.
OK, that might be easier to understand. But, for instance, the holomorphic functional calculus allows you to say that you can take power series of operators and their spectrum behaves like you would expect it to.
I vaguely remember it being useful when working with resolvents of compact operators.
But I think I told you before I'm hardly expert at this.
@MikeMiller I just want an idea of the importance. You are referring to result which states: If $f \in \text{Hol}(\sigma(a))$ then $f(z) = \sum c_k(z- \lambda)^k$ and $f(a) = \sum c_k(a- \lambda)^k$.
What do you mean their spectrum behaves like you want it to.?
I want to ask a question about math PhDs in Europe. Is it obligatory to teach in the PhD life? Or is there any requirement to correct students' answers?
I only remember that I took a course in functional analysis the last year and it contained some kind of continuous functional analysis to prove spectral theorem.
Now I'm still taking courses, but the next semester I will finish a thesis, otherwise I cannot graduate.
I never work in analysis. I just meant that this year, as far as I foresee, I will take no course in analysis and what I'm studying is far away from analysis.
my french is a little rusty. in the following, are they saying that $X$ is demanded to be finite, or that the assumption is guaranteed when $X$ is finite?
6) pour tout $x \in X$, $\rho(x)$ est une immersion (ce qui est une conséquence de a) si K est de caractéristique 0 et $X$ de dimension finie.
@FrankScience Don't worry too much about it, I would imagine that if you talk to your prof about that you can avoid teaching for a while until you've honed your English skills sufficiently
@Danu thanks for the comment, I am in $\mathbb R^2$ ;), here Fourier transforming massless boson propagators $1/p^2$ gives some problems for some reason that cannot be normalised away, to understand that I wanted to see what the Fourier transform of $1/(p^2+m^2)$ looks lie
@BalarkaSen Hmm, I found this very nonintuitive. In fact, when I saw that question, I immediately think of that homeomorphism mentioned in the bottom paragraph, and then the paragraph told me that this $f$ is actually kinda piecemeal function. (because $q\in Q$ is split into 3 cases). I never have any thoughts that resemble the top paragraph, perhaps I might be overlooking something
@BalarkaSen Hmm ,let me think, is it because if I attempt to just deform the letter Q into that O< thing I must have one segment to pass through the loop and this is where the f will fail to be bijective for the case where homeomorphism of the ambient $\mathbb{R}^2$ is wanted?
@BalarkaSen So, in blowing up along a linear subspace $\Bbb C^m\subset \Bbb C^{n+1}$, is there a nice way to see that $\sigma^{-1}(\Bbb C^m)=\Bbb P(\mathcal N_{\Bbb C^m/\Bbb C^{n+1}})$, where $\sigma$ is the projection of the blowup to $\Bbb C^{n+1}$?
In my pictures, I'm just drawing $\Bbb R^2$ with $\Bbb R=\{y=0\}$ as the subspace
Now note that you don't need all of the line. Just a direction, because it's a local construction. Line through 0 in T_0 C^{n+1}, in the tangent space.
@TobiasKildetoft which one is the standard spivak topology text? https://www.amazon.com/s?ie=UTF8&page=1&rh=n%3A226700%2Cp_lbr_one_browse-bin%3AMichael%20Spivak
The only MSE that pop in in google mention about it, but it only say it is a standard topology text. I presume it is well known. The problem is I don't know about it thus without the name I cannot find it
1/2 + 1/3. We need to (...) to write it like this: 3/6 + 2/6 = 5/6 (in French we would say "put to the same denominator" but this is non correct in english)
