@fpqc Looks like you study algebraic geometry. That's one of the things I know nothing about (except the ez things I learnt from a month of reading Shafarevich part I). :P
In order for a limit to exist, you need that the left and right limit exist, and that they equal eachother and that they equal the value of the function
I am trying to prove the Alaoglu's Theorem. But my professor told me there is something wrong with my proof. Can anyone help me? Thank you!
The Alaoglu's Theorem: Let $X$ be a normed space. Then ball $X^*$ is weak-star compact.
Proof:Let $D_x=\{\alpha\in\mathbb{F}:|\alpha|\leqslant1\}$ for each...
The value of the function at zero is irrelevant to whether the limit at 0 exists. (at least according to the definition of limit taught in the US. The french have a different definition, I think?)
@Semiclassical I think that the french (and hence belgians) sometimes define the limit as just "left and right limit exist and equal", without caring about the value of the function at that point. At least that's how I learned it in school.
Well, maybe they stick to that other definition in France then. I learned the french definition in school, but in university we saw the "international" definition.
@BalarkaSen If you care about topology, maybe you'll care about \'{e}tale cohomology.
Basically Andre Weil realized that if we could have a "Weil Cohomology Theory", i.e. something that behaves like singular cohomology for varieties over a finite field
@BalarkaSen We would be able to prove the functional equation and rationality of zeta
@BalarkaSen The problem with learning etale cohomolgoy is that you need to know algebraic geometry. The proofs are often quite complicated and involve a lot of devissage
@BalarkaSen At some point there's various annoying commutative diagrams that one has to check are true. And no one knows except Brian, de Jong and Gabber
Guys, this is a proof I’ve never been able to grasp properly. So I’m guessing they mean that $$ \left\vert \frac{(g\circ f)(x_n)-(g\circ f)(a)}{x_n-a} \right\vert= \left\vert \frac{g(y)-g(f(a))}{y-f(a)} \right\vert\cdot \left\vert \frac{f(x_n)-f(a)}{x_n-a} \right\vert. $$ I don't really understand why they take $y\in I\setminus\{a\}$, instead of $y_n$? I understand that they kind of want to avoid dividing by $y_n-f(a)=0$, but why are we allowed to write the above then?
I am trying to prove the Alaoglu's Theorem. But my professor told me there is something wrong with my proof. Can anyone help me? Thank you!
The Alaoglu's Theorem: Let $X$ be a normed space. Then ball $X^*$ is weak-star compact.
Proof:Let $D_x=\{\alpha\in\mathbb{F}:|\alpha|\leqslant1\}$ for each...
