3
votes
0answers
72 views

Is there a gap in Serre's proof of inverse function theorem?

On page 73 of 'Lie algebras and Lie groups', Serre proves the inverse function theorem for complete fields. I would like to have some clarification about the following point. Let $K$ be a complete ...
5
votes
2answers
172 views

equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous ...
1
vote
0answers
44 views

What is a one-parameter global Lie group of smooth diffeomorphisms?

consider the one-parameter map acting on $\mathbb{R}^2 (t\in\mathbb{R})$: $f:(t,(x_1,x_2))\mapsto \left( x_1+t, \frac{x_1 x_2}{x_1+t} \right)$. I need to prove, that the function $f$ defines a ...
3
votes
1answer
171 views

Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)

If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
6
votes
1answer
321 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
2
votes
0answers
208 views

What is the constant $e$, fundamentally? [duplicate]

Possible Duplicate: Why is the number e so important in mathematics? Intuitive Understanding of the constant “e” The number $e$ is important in many respects. If you ask ...
0
votes
1answer
237 views

Lie algebra of the bounded continuous functions

I can think of the set of bounded, continuous functions from R -> R as a group, with composition as addition of functions. {In other words, this group has the rule that the composition of two ...