Why is (x,e_i) a zero sequence?

In summary, an infinite orthonormal system is considered closed if the sum of the squares of its components is equal to the square of the norm of any element in the system. This is possible because the sequence of the components of the system approaches zero as the index approaches infinity, which can be proven using the nth term test.
  • #1
mathmari
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Hey! :eek:

An infinite orthonormal system $\{e_1, e_2, ... \} \subset H$ is closed in $H$ iff $\forall x \in H$
$$||x||^2=\sum_{i=1}^{n}{|(x,e_i)|^2}$$

From the summability of the right part of the relation above, we conclude to that the sequence $(x,e_i)$ is a zero sequence.

Could you explain me how we conlude to that?
 
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  • #2
Do you mean why $(x,e_i)\to0$ as $i\to\infty$? This follows from the nth term test.
 
  • #3
Evgeny.Makarov said:
Do you mean why $(x,e_i)\to0$ as $i\to\infty$? This follows from the nth term test.

Aha! I got it! Thank you! (Smirk)
 

FAQ: Why is (x,e_i) a zero sequence?

1. Why is (x,e_i) a zero sequence?

There are several possible reasons why (x,e_i) may be considered a zero sequence. One possibility is that the value of x approaches zero as i increases. Another possibility is that the limit of (x,e_i) as i approaches infinity is equal to zero. It is also possible that the product of x and e_i approaches zero as i increases.

2. What is a zero sequence?

A zero sequence is a sequence of numbers or functions that approaches zero as the index or variable increases without bound. In other words, the values of the sequence get closer and closer to zero, but never actually reach zero.

3. How do you determine if (x,e_i) is a zero sequence?

To determine if (x,e_i) is a zero sequence, you can take the limit of the sequence as i approaches infinity. If the limit is equal to zero, then (x,e_i) is a zero sequence. Another way to determine if a sequence is a zero sequence is to check if the product of the terms in the sequence approaches zero as the index increases.

4. What are some examples of zero sequences?

Some examples of zero sequences include (1/i), (1/(i^2)), and (1/i!). These sequences all approach zero as the index i increases without bound.

5. How are zero sequences used in mathematics?

Zero sequences are used in various areas of mathematics, including calculus, analysis, and number theory. They are often used to prove the convergence of series and integrals, as well as to show the existence of limits and derivatives. Zero sequences are also important in studying the behavior of functions and in understanding the concept of infinity.

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