Mykola Pochekai

Does anybody have a good references for fact that stable homotopy groups forms generalized homology theory (in sense of Eilenberg–Steenrod axioms)?

See ya! Thanks for helping and good night! :3

Ok, thanks! I think that your first argument: "You can define $f$ on the subspace $E\oplus[v]$ first and then extend it using Hahn-Banach theorem.", - much simplier than argument on your book.

Truly...

It is always possible to choose nonzero functional $f$ such that $f|_E = 0$ and $E \subset F$ - closed subspace?

Ok, but I like to think about that problem, I always find something new in trying to solve it. :3 So, I will continue to try.

Ok, good luck on your deals, and thanks for your helping! It is really useful for me!

Oh yes... I knew that injective and topological injective it is not the same, but forget it ):

Yes, thats it.

Or T is injective and exists some constants c, C > 0 that c ||x|| \leqslant ||Tx|| \leqslant C ||x|| (it is equivalent)

T - topological injective iff T - homeomorphism between dom T and Im T (in topological sense)

Ok. Hence, it is inaccuracy in the Helemskii textbook.

Yes I also notice that, but I think that I am too stupid to understand that is the same.

Its ok :3 Do you find remark?

Yes but at bottom there must be remark: "Statement (ii) true in two sides, but proof of that difficult, see <refference on Rudin>" or something like that.

It is fast, 2 minutes.

Ok, but I read it in russian language, so I need to download english version to reffer page

I will screen your a page that confused me.

amazon.com/…

I think it is error in Khelemsky book.

I am too, wait a few minutes...

Aw, yes. I think that T(E) may be not closed is problem. I found proof of that fact in Rudin "Functional analysis" theorem 4.15 but he use more powerful methods.

Just a minute, I think about your proof.

Sorry. I confused problems. Right problem is that: "if $T^* : F^* \to E^*$ is injective than $T : E \to F$ surjective".

Banach spaces. T - linear bounded operator. * - adjoint.

@MartinSleziak For me too :3 i wanna to find naturally proof of next fact: "if $T^* : F^* \to E^*$ surjective then $T : E \to F$ injective". Proof of that fact are exist, but it use very difficult techniques. I think naturally proof can be finded if we find some good properties of banach adjoint functor. I tried proof that fact the third week in a row 3:

Hi there! I am interesting about homological methods in functional analysis, maybe someone else interesting too?