How to find the distance between these elements?

[futurebird]

I'm studying the proof that the l₂ metric space is complete and I just don't get something about this proof.

What I know:

1. l₂ is the metric space whose elements are sequences that converge when you square each term and sum them. In other words, a sequence, f is in l₂ if SUM( |f(k)|²) < infinity.
2. The norm on this metric space is: (SUM( |f(k)|²))^0.5 = || f ||₂
3. A sequence in the l₂ space is a sequence of sequences (confusing isn't it?)
4. To show that the space is complete I need to show that every cauchy sequence in l₂ converges to a sequence in l₂.

My book says:

Suppose that for e > o there exists n₀ such that:

||fn - fm||₂ < e when m, n >= n₀.

I don't understand how to think about ||fn - fm||₂.

Suppose fn = .1, .01, .001, ...
Suppose fm = .2, .02, .002, ...

How would I find ||fn - fm||₂ computationally?

Do I do:

||fn - fm||₂ = (|(.1² + .01², .001² + ...) - (.2² + .02² + .002² + ...)|)^0.5
||fn - fm||₂ = (|(.01 + .001, .0001 + ...) - (.04+ .004 + .0004 + ...)|)^0.5
||fn - fm||₂ = (|(.01 + .001, .0001 + ...) - (.04 + .004 + .0004 + ...)|)^0.5
||fn - fm||₂ = ( |1/90 - 4/90|)^0.5
||fn - fm||₂ = (3/90)^0.5

or

[(.1 - .2)² + (.01-.02)², + ...]^0.5 ?

I think it's the first way... but for some reason I'm confused. Deeply.

 

[Dick]

It's the second way. First find the difference between the sequences, then calculate the norm. Your first calculation is | ||fn|| - ||fm|| |.

 

[futurebird]

Thanks dick... OK... So...

|| fn - fm||2 = [(.1 - .2)^2 + (.01-.02)^2 + ...]^0.5
|| fn - fm||2 = [(.1)^2 + (.01)^2 + ...]^0.5
|| fn - fm||2 = [.01 + .001 + ...]^0.5
|| fn - fm||2 = [1/90]^0.5

1 / sqrt(90) is about 0.105409255Grrr... Ok so:

|fn(k) - fm(k)| <= || fn - fm||2

|fn(k) - fm(k)| is the absolute difference between the kth terms in fn and fm ... sequences that are "close enough together" so that || fn - fm||2 < e.

Suppose

fn = .1, .01, .001, ...
fm = .2, .02, .002, ...
are that close together. This is saying that

.1 < = || fn - fm||2

.1 < = 0.105409255

Yay! it worked! I don't know why but it did!

Thanks!

God... but why is |fn(k) - fm(k)| <= || fn - fm||2?

I'm not quite seeing that...

 

[futurebird]

Oh wait I think I see that now... it's like:

x-y <= sqrt((x-y)^2 + positive stuff or zero)I wish I knew what this was all about... It seems very disconnected from the other things in the course... I just don't know where it's going ...

 

[Dick]

Oh wait I think I see that now... it's like:

x-y <= sqrt((x-y)^2 + positive stuff or zero)


I wish I knew what this was all about... It seems very disconnected from the other things in the course... I just don't know where it's going ...

You mean |x-y|. Yes, that's what it's like. BTW I get 1/sqrt(99) for your first example for ||fn-fm||. The common ratio in the geometric series is wrong. It's 1/100, not 1/10.

 

[futurebird]

You mean |x-y|. Yes, that's what it's like. BTW I get 1/sqrt(99) for your first example for ||fn-fm||. The common ratio in the geometric series is wrong. It's 1/100, not 1/10.

ah yes.. see that now. OK... this proof is starting to make a little more sense to me now. Thank you!