Proof of convergence (intro topology)

In summary, we are trying to prove the convergence of the series \sum^{\infty}_{i=1} |x_{i}y_{i}|, given that x = (x1, x2,...) and y = (y1, y2,...) are members of l^2, which is defined as { x=(x_{1}, x_{2}, ... ) \in ℝ^{\omega} : \sum^{\infty}_{i=1} (x_{i})^{2} converges }. We use the Cauchy-Schwarz inequality and algebraic manipulation to prove the convergence of the series.
  • #1
1MileCrash
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Homework Statement



Show that if x = (x1, x2,...) and y = (y1, y2,...) are members of l^2, then

[itex]\sum^{\infty}_{i=1} |x_{i}y_{i}|[/itex]

Converges

Homework Equations



My book defines l^2 to be:

[itex]{ x=(x_{1}, x_{2}, ... ) \in ℝ^{\omega} : \sum^{\infty}_{i=1} (x_{i})^{2} converges }[/itex]

(should be set brackets around that, don't know why they don't show up.)

The Attempt at a Solution



Concerns:

-I have no idea if Cauchy-Schwarts inequality actually holds in an infinite dimensional space, and can't find any information regarding whether it does or not.

-I can't think of any way to relate my last line to the expression I am showing convergence for.

-I have no idea if just because two series are convergent, then their product is convergent. I assumed it was anyway.

______________

Proof:

Since x and y are both members of l^2,

[itex] \sum^{\infty}_{i=1} (x_{i})^{2}[/itex]

and
[itex]\sum^{\infty}_{i=1} (y_{i})^{2}[/itex]

converge.

Then, the root of these series must converge too. The roots of these series are the norms of x and y.

So, ||x|| and ||y|| converge.

The Cauchy-Schwarts inequality states that

[itex]|x \cdot y| \leq ||x|| ||y||[/itex]


Since ||x|| ||y|| converges, so does |x (dot) y|

SCRATCH that!

I thought that was so because of the comparison test for convergent series, but the comparison test requires a comparison to be made between the terms of the series, not the series themselves...

So, I'm at square one.

Any help with this one?
 
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  • #2
hmm I think trying first to prove the convergence of (x + y)^2 would be a good start..
 
  • #3
Could you shed some light on the three concerns I've outlined? I'm really shooting around in the dark here, I don't know how to show (x+y)^2 converges. I've never done a single proof regarding convergence before.
 
  • #4
No ideas?
 
  • #5
Use the inequality ##2ab \leq a^2 + b^2## for real numbers ##a## and ##b##.

That said, Cauchy-Schwarz is valid for infinite dimensional space such as ##\ell^2##, but then you need to prove it.
 
  • #6
Hmm OK, I see where you're going with that. I'm away from all my books now, but is going to look like:

Square each series, it converges (because they are absolutely convergent?) add the two convergent series, then due to your inequality I can use the comparison test and confirm that 2xy converges, then so does xy, and the series xy is exactly the dot product?
 
  • #7
Why do you use absolute convergence? You don't know that your series converge absolutely.
 
  • #8
I don't, but I wouldn't know how to proceed if they aren't. What I've read says that if a is convergent and b is convergent, I can only say that ab is convergent if a or b converges absolutely.
 
  • #9
You know ##\sum|x_n|^2## converges. Isn't that enough?
 
  • #10
Ohh, yes it is. I forgot that exponent was in the l^2 criteria.
 
  • #11
Okay, now I've learned a lot. From now one, when I'm attempting to prove convergence, I'll try to look for an algebraic identity that allows me to compare first.

Since x and y are members of l^2,

[itex]\sum^{\infty}_{i=1} (x_{i})^{2}[/itex]
[itex]\sum^{\infty}_{i=1} (y_{i})^{2}[/itex]

both converge.

Since members of l^2 are points in R^omega, we know that xi and yi are real numbers, so

[itex]\sum^{\infty}_{i=1} |x_{i}|^{2}[/itex]
[itex]\sum^{\infty}_{i=1} |y_{i}|^{2}[/itex]

both converge, since k^2 = |k|^2 for any real.

The sum of two convergent series also converges.

[itex]\sum^{\infty}_{i=1} |x_{i}|^{2} + |y_{i}|^{2}[/itex]

Consider that

[itex](a-b)^{2} \geq 0[/itex]
[itex]\Rightarrow a^{2} - 2ab + b^{2} \geq 0[/itex]
[itex]\Rightarrow a^{2} + b^{2} \geq 2ab[/itex].

This inequality let's us say that

[itex]|x_{i}|^{2} + |y_{i}|^{2} \geq 2|x_{i}||y_{i}|[/itex]

Then, by the comparison convergence test,

[itex]\sum^{\infty}_{i=1} 2|x_{i}||y_{i}|[/itex]

converges.

Since a convergent series multiplied by a constant is still convergent, and |n||m| = |nm|, we can say that

[itex]\sum^{\infty}_{i=1} |x_{i}y_{i}|[/itex]

converges, which was the goal.
 
  • #13
Thanks again
 
  • #14
I've been trying to find more info on this l^2 set but googling doesn't work very well. Does it have another name?

I'm wondering if it is defined this way because it makes many metrics and the operation above (which is not a metric, but is related to a basic operation) work.
 

1. What is proof of convergence in topology?

Proof of convergence in topology is a mathematical concept that is used to show that a sequence of points in a topological space converges to a specific limit. This proof typically involves showing that as the terms in the sequence get closer and closer to the limit, they eventually get arbitrarily close, or within a given distance, to the limit.

2. How is proof of convergence different from convergence in other areas of mathematics?

In other areas of mathematics, convergence is typically defined in terms of a sequence of real numbers or functions. However, in topology, convergence is defined in terms of a sequence of points in a topological space, which may not necessarily be numerical values.

3. What are the main tools used in proof of convergence in topology?

The main tools used in proof of convergence in topology include the definition of a topological space, the concept of a limit point, and the use of open sets and neighborhoods to show that a sequence of points converges to a given limit.

4. Can a sequence have multiple limits in topology?

Yes, in topology, a sequence can have multiple limits. This is because the concept of convergence in topology is not based on a specific value, but rather on the behavior of the sequence as it approaches a given point in a topological space. If a sequence has multiple limits, it means that it converges to different points in the topological space.

5. How is proof of convergence used in practical applications?

Proof of convergence is used in practical applications, such as computer algorithms and optimization problems, to ensure that a sequence of points is approaching a desired solution or limit. It is also used in physics, engineering, and other fields to analyze and predict the behavior of systems over time.

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