Proving <x, T_a> is Nonzero for Countable a in Hilbert Space H

In summary: So, what inequality did you mean?In summary, we are trying to show that the inner product <x, T_a> is different from 0 for at most countably many a. This can be proven by considering the sum of |<x, T_a>|^2, which does not converge for an uncountable number of terms. By using the Bessel inequality and the fact that the sum of <x, T_a>^2 is less than or equal to the sum of |<x, T_a>|^2, we can conclude that there are countably many a for which <x, T_a> is greater than a fixed positive number e. This is because if there were an uncountable number
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{T_a} is an orthonormal system (not necessarily countable) in a Hilbert space H. x is an arbitrary vector in H.


i must show that the inner product <x, T_a> is different fron 0 for at most countably many a.

i'm not even quite sure where to begin. i know that the inner product is the sum(B_aT_a) in a hilbert space, but I'm sure what other information i have to work with.

thanks
 
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  • #2
Can you sum an uncountable number of non-zero terms?
 
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And what can you say about the sum of |<x,T_a>|^2?
 
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could i use the Bessel inequality?
 
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or if i fix an e>0...for how many a can it be true that the absolute value of <x, T_a> is greater than e?
 
  • #6
Take the hints you were given. It is impossible for this vector to have a length because you cannot sum an uncountable number of strictly positive terms. Prove it.
 
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Raven2816 said:
or if i fix an e>0...for how many a can it be true that the absolute value of <x, T_a> is greater than e?

This is on the right track. Now answer your own question. How many? Be careful though - how do you know sum |<x,T_a>| converges? Hint: in general, it doesn't.
 
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that's a great question. how DO i sum the absolute value of inner products when there are infinite terms? it won't converge. so for a fixed e>0, there are countably many a for which <x, T_a> is greater than e...but why
 
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What does Bessel's inequality tell you? As for how many terms in a convergent infinite series can be greater than some positive number e, can the number even be infinite?
 
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Raven2816 said:
that's a great question. how DO i sum the absolute value of inner products when there are infinite terms? it won't converge. so for a fixed e>0, there are countably many a for which <x, T_a> is greater than e...but why
You learned in calculus how to sum an infinite number of terms! But those were all countable sums. Precisely what is meant by "linear combination" here? If there were an uncountable number of terms with non-zero inner product that would give a linear combination of the basis vectors with an uncountable number of non-zero terms. Is that possible the way "linear combination" is defined in a Hilbert space?
 
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i know there are countably many nonzero terms...because eventually i will have terms that are no longer linearly independent.
 
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i know there are countably many a such that for e>0 the absolute value of <x, T_a> is greater than e. i know that i can't sum an uncountable number of strictly positive terms because it doesn't converge. i just don't understand everything in the middle that should tie this together.
 
  • #13
I don't follow you.

You claim that you can show that if <x,T_a> is non-zero for uncountably many a (which implies <x,x> is infinite), then it is a contradiction. But that was all you were asked to show.
 
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i know that <x, T_a> is non-zero for countably many a. I'm just struggling to show that this is in fact true.
 
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Um. But you just said that if it were false, then you can show a contradiction. Therefore it is true. You can't sum an uncountable number of positive terms. And you know

<x,x>^2 => sum <x,T_a>^2

and the RHS doesn't exist if there are not countably many non-zero terms.
 
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so i can say that since that equality is true, and i can't sum uncountable positive terms, that obviously there are countably many a where the inner product is nonzero? is that a complete answer? it makes intuitive sense, but is it legit?
 
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Of course it is legit. Why wouldn't it be? You've shown not(B) implies not(A), thus A implies B.
 
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Raven2816 said:
so i can say that since that equality is true, and i can't sum uncountable positive terms, that obviously there are countably many a where the inner product is nonzero? is that a complete answer? it makes intuitive sense, but is it legit?

WHAT equality is true? I haven't seen any equalities playing a big role here. And the reason why you can't have an uncountable number of positive terms in a convergent series isn't because it's 'obvious'. I'm not even sure any of this is 'intuitive'. I think you are going to have to state more of this in the form of a 'proof'.
 
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I think he meant inequality.
 
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matt grime said:
I think he meant inequality.

I hope so.
 

1. What is a Hilbert Space?

A Hilbert Space is a mathematical concept that represents an infinite-dimensional vector space. It is named after the German mathematician David Hilbert and is commonly used in functional analysis and quantum mechanics.

2. What does it mean for to be nonzero?

In Hilbert Space, represents the inner product between a vector x and a linear operator T_a. A nonzero value indicates that the vector x and operator T_a are not orthogonal and have a non-trivial relationship.

3. How do you prove is nonzero?

To prove is nonzero, you can use the definition of the inner product to show that it is not equal to zero. This can be done by showing that either the vector x or the operator T_a is not equal to zero, or by using other mathematical techniques such as the Cauchy-Schwarz inequality.

4. What does it mean for a to be countable in the context of Hilbert Space?

In Hilbert Space, a countable set refers to a set that can be put into a one-to-one correspondence with the natural numbers. This means that the set can be enumerated, or listed, in a sequence such as 1, 2, 3, ..., n. In the context of proving is nonzero, a countable a indicates that the inner product is being evaluated at a specific, discrete value.

5. Why is proving is nonzero important in the study of Hilbert Space?

Showing that is nonzero is important because it helps establish the existence of a non-trivial relationship between the vector x and operator T_a. This can have implications for understanding the behavior of certain systems and can lead to important insights in functional analysis and quantum mechanics.

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