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This is an experiment. I thought of a way to bridge the gap between the

Now everybody can

Now let's see whether we can prove something!

Edit: Group operation is ##(fg)(x)=f(g(x))## and integration Lebesgue. I also corrected the domain accordingly. Two dimensional might have been more fun, but let's start simple and see who participates.

*usual*challenge threads. Of course we could shorten the monthly period, but given that there are almost always untouched problems, more of them might not be the solution. Today we had a thread "Is math a language" by @frankin garcia and most of us will certainly think so. So if it is, then we can write an essay in this language, formerly known as proof. I have no idea how it will go, where it ends, or if it makes sense at all. I thought we could give it a try until the next load of challenges in December. We start with:**Let**##G##**be a not necessarily Abelian group of square integrable smooth, real functions**##f\, : \,I=[0,1]\longrightarrow [0,1]##**on the unit interval, and**##\mathfrak{g}##**its real Lie algebra.**Now everybody can

**either**add a conclusion based on all previous posts, e.g. "Since ##I## is compact, all functions ..."**or**add additional properties, e.g. "Assume ##G## is simple."**or**focus on additional perspectives, e.g. "Let us consider the center ##Z(G)## of ##G## ...". Conclusions must be proven (keep it short) or the relevant theorem must be quoted. In my example it could be e.g. Heine-Borel or Weierstrass, depending on which direction you want to go: topology or analysis. Please choose only one of these possibilities per post and do not post more than once in a row, i.e. you may continue after somebody else posted something. The projected runtime is until end of month, but it will depend on what actually will happen.Now let's see whether we can prove something!

Edit: Group operation is ##(fg)(x)=f(g(x))## and integration Lebesgue. I also corrected the domain accordingly. Two dimensional might have been more fun, but let's start simple and see who participates.

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