Convergence in Hilbert space question

In summary, it is not possible to show that in a Hilbert space, if z_n is a sequence such that (z_n,y)-->0 for all y, then z_n-->0. This is demonstrated by considering an orthonormal basis of the Hilbert space and showing that the basis vectors do not converge even though their inner products with any vector tend to zero.
  • #1
quasar987
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Homework Statement


Is it true/possible to show that in a Hilbert space, if z_n is a sequence (not known to converge a priori) such that (z_n,y)-->0 for all y, then z_n-->0 ?


The Attempt at a Solution



I've shown that if z_n converges, then it must be to 0. But does it converge?
 
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  • #2
Consider an orthonormal basis of the Hilbert space, {e_n} and let the sequence be the basis vectors. Now (e_n,y) is the nth component of y, which must tend to zero as n goes to infinity for any y. Yet e_n clearly does not converge. So no, you can't show that.
 
  • #3
Good example!
 

Related to Convergence in Hilbert space question

1. What is convergence in Hilbert space?

Convergence in Hilbert space refers to the behavior of a sequence of vectors in a Hilbert space as the number of terms in the sequence increases. It is a measure of how close the terms of the sequence get to a specific limit or target vector.

2. How is convergence in Hilbert space different from other types of convergence?

Hilbert space is a special type of vector space with additional structure, such as a norm and an inner product. This allows for a more precise definition of convergence, compared to other types of convergence in general vector spaces.

3. What is the importance of convergence in Hilbert space?

Convergence in Hilbert space is a fundamental concept in various fields of mathematics, including functional analysis, operator theory, and numerical analysis. It plays a crucial role in understanding the behavior of sequences and series of functions, and is used to prove the existence of solutions to many mathematical problems.

4. What are some examples of sequences that converge in Hilbert space?

Some common examples of sequences that converge in Hilbert space include the sequence of partial sums of a convergent series, the sequence of Fourier coefficients of a square-integrable function, and the sequence of eigenvalues of a compact operator.

5. How is convergence in Hilbert space related to the concept of completeness?

In Hilbert space, a sequence converges if and only if it is a Cauchy sequence, meaning that the terms of the sequence get arbitrarily close to each other. This is equivalent to the space being complete, which means that every Cauchy sequence in the space converges to a limit within the space. Therefore, convergence in Hilbert space is intimately related to the concept of completeness.

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