Hilbert Spaces Proof

Mar 2010
19
0
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
 

Drexel28

MHF Hall of Honor
Nov 2009
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Berkeley, California
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
What have you tried? The second one is a famous result and can easily be found by googling it.

For the first one are we to assume that the sequences are in \(\displaystyle \mathbb{R}\)? For this specific space isn't the mapping \(\displaystyle \eta:X\to\mathbb{R}:\bold{x}\mapsto\sum_{n=1}^{\infty}\frac{\pi_n(\bold{x})}{n}\) continuous and \(\displaystyle M=\eta^{-1}(\{0\})\)?
 
Mar 2010
19
0
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
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
First one I have no idea what im doing, the sequence are on the complex plane C.
Well, my question still holds. If \(\displaystyle \eta\)'s codomain is made into \(\displaystyle \mathbb{C}\) is everything I said still correct?

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
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 \(\displaystyle \text{bounded }\Leftrightarrow\text{ totally bounded}\) 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.

So, I would hold out for a more knowledgeable member, like Opalg or somebody.
 

Jose27

MHF Hall of Honor
Apr 2009
721
310
México
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
I don't get the first one at all. \(\displaystyle X\) as defined is not even a vector space (the sequences \(\displaystyle x_n=1,y_n=-1\) are in the space but \(\displaystyle x_n+y_n=0\) 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 \(\displaystyle \hat{x}\) the image of \(\displaystyle x\) under this mapping then:

\(\displaystyle \sup_{\| \ell \| =1} |\ell (x) | = \sup_{ \| \ell \| =1} |\hat{x} (\ell ) | = \| \hat{x} \| = \| x\|
\)

Now just use the Riesz representation theorem.

Edit: \(\displaystyle \ell\) is, of course, in the dual space.
 
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Mar 2010
19
0
I don't get the first one at all. \(\displaystyle X\) as defined is not even a vector space (the sequences \(\displaystyle x_n=1,y_n=-1\) are in the space but \(\displaystyle x_n+y_n=0\) 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 \(\displaystyle \hat{x}\) the image of \(\displaystyle x\) under this mapping then:

\(\displaystyle \sup_{\| \ell \| =1} |\ell (x) | = \sup_{ \| \ell \| =1} |\hat{x} (\ell ) | = \| \hat{x} \| = \| x\|
\)

Now just use the Riesz representation theorem.

Edit: \(\displaystyle \ell\) 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 !
 

Jose27

MHF Hall of Honor
Apr 2009
721
310
México
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, \(\displaystyle X\) 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 \(\displaystyle X= \{ (x_n)\in \ell ^2 (\mathbb{N} ) : x_n \neq 0 \ \mbox{for finitely many} \ n \}\)?

For the third one, have you seen the Hahn-Banach theorem, because then just define \(\displaystyle \ell \left( \frac{x}{\| x \| } \right) =1\) and extend to the whole space.
 
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Mar 2010
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Okay, what is your definition of the orthogonal of a subspace, since as I showed, \(\displaystyle X\) 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 \(\displaystyle X= \{ (x_n)\in \ell ^2 (\mathbb{N} ) : x_n \neq 0 \ \mbox{for finitely many} \ n \}\)?

For the third one, have you seen the Hahn-Banach theorem, because then just define \(\displaystyle \ell \left( \frac{x}{\| x \| } \right) =1\) and extend to the whole space.
Oh sorry your right. The norm im using is l(squared) !

And no I haven't heard of Hahn-Banach theorem. We only have started linear operators if that helps.

Thanks for the link to question 2, I've got that all sorted
 

Jose27

MHF Hall of Honor
Apr 2009
721
310
México
Oh sorry your right. The norm im using is l(squared) !

And no I haven't heard of Hahn-Banach theorem. We only have started linear operators if that helps.

Thanks for the link to question 2, I've got that all sorted
If \(\displaystyle X\) is the space I gave, then Drexel's argument gives you the answer to the first part, for the second, given \(\displaystyle (y_n) \in X\) consider \(\displaystyle \hat{y} _n := ny_n\) and if \(\displaystyle N> \max_{m} \{ y_m \neq 0\}\) is fixed \(\displaystyle \hat{y} _N := -N\sum_{n} y_n\). Prove \(\displaystyle (\hat{y} _n) \in M\), what happens if \(\displaystyle \sum \overline{y} _n\hat{y} _n =0\)?

For the third, define \(\displaystyle \ell (x)= \| x\|\) and \(\displaystyle 0\) in the orthogonal complement of the subspace generated by \(\displaystyle x\). Prove it's continous and of norm \(\displaystyle 1\)
 
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Mar 2010
19
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If \(\displaystyle X\) is the space I gave, then Drexel's argument gives you the answer to the first part, for the second, given \(\displaystyle (y_n) \in X\) consider \(\displaystyle \hat{y} _n := ny_n\) and if \(\displaystyle N> \max_{m} \{ y_m \neq 0\}\) is fixed \(\displaystyle \hat{y} _N := -N\sum_{n} y_n\). Prove \(\displaystyle (\hat{y} _n) \in M\), what happens if \(\displaystyle \sum \overline{y} _n\hat{y} _n =0\)?

For the third, define \(\displaystyle \ell (x)= \| x\|\) and \(\displaystyle 0\) in the orthogonal complement of the subspace generated by \(\displaystyle x\). Prove it's continous and of norm \(\displaystyle 1\)
Ok question 1 is still so confusing, but I'll give it another try, I don't really get what your trying to do.


Thanks for the help with the other ones though