Hey all, thanks for the replies and comments and I apologize for getting back a little late.
Not really sure why I threw that statement in there, there are plenty of counter-examples.
I appreciate the help, thanks!
1. Homework Statement
I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the...
(Edit: I realize I am relying heavily on Lang's prior proof of the existence of the free group (apparently owed to J. Tits). Please let me know if I should fill it out in detail here if it is not readily accessible to you via the internet or by the text.)
I had some time to think about this...
The DCT implies that , ##\lim_{n\to\infty}\int_{0}^{1}f_n d\mu=\int_{0}^{1}gd\mu## for a sequence ##f_n## that converges pointwise to ##g##. All you have is a pointwise supremum, ##g=\sup_n f_n##, which is easily seen to be measurable. In any case, I can't see how the DCT would imply here that...
I think I misinterpreted your question the first time around. Do you mean by 'the normal subgroup generated by ##T##' the normal closure of ##T##? The normal closure of ##T## is the minimal normal subgroup containing ##T##.
If so, then let's claim that...
As a Fourier series, the first term is ##\frac{A^2/2}{2}=\frac{A^2}{4}## (i.e., divided by ##2##). That, I believe, is what's going on here.
EDIT (Again D: )
OK! My bad.
What's happening here is this. The sum converges, so we rearrange a finite number of terms (i.e., one term) and notice that...
If by this, you mean that ##\langle gTg^{-1}\rangle=\langle{T}\rangle## for any subset of a group, then I dispute your claim.
Just consider a group ##G## where ##G'## is a subgroup of ##G## that is not normal (i.e., ##gG'g^{-1}\ne G##). Indeed, if the set ##\{g_1,\ldots,g_n\}## generate...
Homework Statement
Coproducts exist in Grp. This starts on page 71. of his Algebra.
Homework Equations
[/B]
Allow me to present the proof in it's entirety, modified only where it's convenient or necessary for TeXing it. I've underlined areas where I have issues and bold bracketed off my...
If the negative sign is outside the power, you can pull it out all together until you're done simplifying. If the negative sign is being raised to a power as well, you must bring it along.
So, if you have the negative signs all out front, and not being raised to a power, then yes, because...
(-2)^{5/2}+(-2)^{5/2}=2(-2)^{5/2}
It might help to think about it like this. We write x=y^{a/b} to mean roughly that x is the number equal to y multiplied together with itself "a/b times". In this sense, how might you think we should write 2(-2)^{5/2}?
A couple of notes first:
1.
\hom_{A}(-,N) is the left-exact functor I'm referring to; Lang gives an exercise in the section preceeding to show this.
2.
This might be my own idiosyncrasy but I write TFDC to mean 'The following diagram commutes'
3.
Titles are short, so I know that the hom-functor...