Originally Posted by

**Stranger**
Thanks very much both of you!

I'm sorry I couldn't reply earlier; my computer keeps crashing down. I guess it's time for me to buy a laptop. But these aren't perfect either... whatever, if I don't reply immediately, it only means I'm unable to reply because of that dumb pc.

These problems are not homeworks by the way. It's just that I feel bad when I've tried hard to find a solution and then, not only reach nothing, but also never know the solution...

ok, here is another one. I've hesitated for a long time to post that one because I really feel it's an easy one, but I keep trying and don't reach the thing.

It's problem 22 the same chapter.

22. Let M be a closed subspace of X. Define the quotient space

X/M to be the vector space consisting of the cosets [x] =

{x+z : z € M} with vector addition and scalar multiplication

defined by [x] + [y] = [x+y], a[x] = [ax]. Define ||[x]||_1 =

d(x,M), the distance from x to M. Prove that ||.||_1 is a

norm on X/M and that (X/M,||.||_1 ) is complete.

(d(x,M) is of course the infimum of all d(x,y) for y in M)

I've proved the norm, but I'm a bit stuck with completeness. As usual I assumed {[x]n}n was a Cauchy sequence. Then I finally reached that given eps>0,

d(xn-xm,M)<eps for all n,m>n0. _____(*)

where [xn]=[x]n

Then I claimed that this implies (xn-xm) is in the closure of M (in which case the proof would be finished: M is closed, thus (xn-xm) is in M, thus [xn-xm]=0, thus [xn]=[xm] for all n,m>n0, thus [x]n is eventually stationary, thus convergent)

It turns out I was wrong however, my prof says (*) **does not** imply that (xn-xm) is in the closure.

Any little hint?

Oh, and another thing. In a problem I reached the conclusion that 2 vectors x,y must satisfy:

Re <x,y> = 0

where <x,y> is their inner product. Can I translate this condition into something else on x and y themselves, instead of their product? Or is it all I can say?

Thanks a lot