1) X is a sequence of terminating sequence over C
and let:
oh forgot to type sum{x(n)/n}=0
Show M is a closed subspace of X, and M(perp)={0}
2) Let H be a Hilbert space and let {h(j)} where j belongs to J be a basis. The dimension of H is the cardinality of J .
Show that H is finite dimensional if and only if the unit ball is compact.
3) Let H be a Hilbert space. Show that for any h belong to H
I'm really stuck on 1 and 2. But question 3 I can show the sup is =<1 using C-S equation with llgll=1.
Thanks for help guys
First one I have no idea what im doing, the sequence are on the complex plane C. I tried googling the second one and to no success, if you can link me it would be great. The third one I can get halfway using C-S inequality and showing that the sup is =< 1 but thats only half of it =s
Well, my question still holds. If 's codomain is made into is everything I said still correct?
I frankly cannot help much more than prompt you to search on the internet. I think I recall hearing a theorem that a subset of a normed vector space has the property that if and only if the vector space is finite dimensional. From there your problem would follow fairly easily since a metric space is compact if and only if it's totally bounded and complete.I tried googling the second one and to no success, if you can link me it would be great. The third one I can get halfway using C-S inequality and showing that the sup is =< 1 but thats only half of it =s
So, I would hold out for a more knowledgeable member, like Opalg or somebody.
I don't get the first one at all. as defined is not even a vector space (the sequences are in the space but is not) let alone a Hilbert space for us to talk about an orthogonal subspace. Another thing, in a Hilbert space if a subspace is closed, then we can decompose the space in the direct sum of said subspace and it's orthogonal which gives that the only possible closed subspace with trivial orthogonal is the whole space.
For the second, the difficult part follows from this Riesz's lemma - Wikipedia, the free encyclopedia
For the third one consider the canonical embedding of a space in it's double dual which gives an isometry and call the image of under this mapping then:
Now just use the Riesz representation theorem.
Edit: is, of course, in the dual space.
1) I've had a talk to my lecturer, he said that X is not meant to be complete space. I think I need to show that M is a closed subspace of X first, and then find M(perp). And since X is not complete => M(perp,perp) doesnt equal M
2) Thanks I'll have a look at it now.
3) We haven't started dual spaces yet and I'm sure that the lecturer wants us to use the C-S inequality first
Thanks for the help so far !
Okay, what is your definition of the orthogonal of a subspace, since as I showed, is not even a vector space, so it can't be an inner product space in the usual sense. Just out of curiosity, don't you mean ?
For the third one, have you seen the Hahn-Banach theorem, because then just define and extend to the whole space.
If is the space I gave, then Drexel's argument gives you the answer to the first part, for the second, given consider and if is fixed . Prove , what happens if ?
For the third, define and in the orthogonal complement of the subspace generated by . Prove it's continous and of norm