MHB Why is (x,e_i) a zero sequence?

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Sequence Zero
Click For Summary
An infinite orthonormal system in a Hilbert space is closed if the norm of any vector can be expressed as the sum of the squares of its inner products with the basis elements. This leads to the conclusion that the sequence of inner products, denoted as (x,e_i), is a zero sequence. The convergence of (x,e_i) to zero as i approaches infinity is supported by the nth term test for convergence. The discussion highlights the relationship between the properties of orthonormal systems and the behavior of sequences derived from them. Understanding this connection is crucial for grasping the fundamentals of functional analysis.
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

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?
 
Mathematics news on Phys.org
Do you mean why $(x,e_i)\to0$ as $i\to\infty$? This follows from the nth term test.
 
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)
 

Thread 'Significant figures for inherently bound values'

I have been insisting to my statistics students that for probabilities, the rule is the number of significant figures is the number of digits past the leading zeros or leading nines. For example to give 4 significant figures for a probability: 0.000001234 and 0.99999991234 are the correct number of decimal places. That way the complementary probability can also be given to the same significant figures ( 0.999998766 and 0.00000008766 respectively). More generally if you have a value that...
View full post »

Similar threads

MHB Closed infinite orthonormal system
  • · Replies 2 ·
Replies
2
Views
2K
MHB Identities of the optimal approximation
  • · Replies 9 ·
Replies
9
Views
3K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
I Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
3K
Prove that this series converges uniformly on R
  • · Replies 4 ·
Replies
4
Views
2K
MHB Orthonormal basis - Set of all isometries
  • · Replies 23 ·
Replies
23
Views
2K
MHB Solving a Challenging Number Sequence Problem
  • · Replies 1 ·
Replies
1
Views
1K
Is the Sequence xn=1/(n-2) Convergent for All Values of n?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Gram_Schmidt Orthonormalization .... Remarks by Garling, Section 11.4 .... ....
  • · Replies 6 ·
Replies
6
Views
2K
MHB Every Cauchy sequence converges
  • · Replies 1 ·
Replies
1
Views
2K