Help with Metric Spaces! (Mathematics)

titaniumx3 · 8 Feb 2007 at 21:15

Hi

I'm completely new to metric spaces and I'm having some difficulties getting around the notation and would appreciate some help. In particular I'm stuck on this:

What exactly does the part highlighted in red actually mean? I'm thinking the sup is 'supremum' but to what exactly does that apply? The largest value of |f(x)-g(x)| for any x in [0,1] ?

Also for part (a) I have no clue whatsoever! How would you show the sequence converges to 0?

Any advice would really be appreciated!

Barrington · 8 Feb 2007 at 21:18

Just when i thought i'd gotten away from too much maths at Uni.. here they pop up on the forum

Sorry, can't help you on this one - too complex for me.. it's pretty meaty stuff though :eek:

Psiko · 8 Feb 2007 at 21:45

You pretty much have the sup right, except |f(x) - g(x)| may not have a maximum, so then the sup would be the largest limit point. For example 1-1/x where x is in (0,1) does not have a maximum, but as x->0, 1-1/x -> 1 which is the sup. But, 1 is never actually attained, because it would be attained when x is 1, but 1 is not in (0,1) (this is an open interval.)

To show the sequence f_n converges to g with respect to a distance, show that as n->inf d(f_n,g) -> 0.

Can you see that for d_1, we can always pick a point x in [0,1] such that f_n(x) = 1?

Can you see that the area under this curve will tend to 0 as n->inf?

Feel free to ask if you need any more help.

titaniumx3 · 8 Feb 2007 at 23:11

Thanks that has helped quite a bit, but I still need to get my head around the the convergence of fn and how the distance affects it.

I understand that if fn is to converge to a function g, as n->inf then d(fn, g)->0. So this obviously means fn can only converge to g(x)=0 or g(x)=1, right?

Psiko · 8 Feb 2007 at 23:27

For this question you don't need to think about what it converges to. The question tells you that you are comparing the series of functions f_n to "the zero function." Convergence is all about distance. For one thing to converge to another the distance between them as n->inf must be 0.

titaniumx3 · 8 Feb 2007 at 23:36

Psiko said:
For this question you don't need to think about what it converges to. The question tells you that you are comparing the series of functions f_n to "the zero function." Convergence is all about distance. For one thing to converge to another the distance between them as n->inf must be 0.

Oh ok, that makes sense lol.

So in the case of d_1, as n->inf there always exists an x in [0,1] such that f_n(x)=1, hence where g(x)=0, d_1(f_n,g)=1, i.e. f does not converge to the zero function.

Hope thats on the right lines!

Arcade Fire · 8 Feb 2007 at 23:48

Psiko said:
You pretty much have the sup right, except |f(x) - g(x)| may not have a maximum, so then the sup would be the largest limit point. For example 1-1/x where x is in (0,1) does not have a maximum, but as x->0, 1-1/x -> 1 which is the sup. But, 1 is never actually attained, because it would be attained when x is 1, but 1 is not in (0,1) (this is an open interval.)

Here we're looking at the set of functions continuous on the closed interval [0,1] though, and a continuous function on a closed, bounded interval is both bounded and attains its bounds (a basic result of analysis) so you don't need to worry about that - a supremum is the same as a maximum in this case.

Titanium, what you've said in the above post is fine. Now you just need to think about the distance function d_2. Hint: write out the integral

Int_0^1 |f_n(x)-f(x)| dx

where f is the zero function, and then split the interval into [0,1/n] and [1/n, 1] and go from there.

Psiko · 8 Feb 2007 at 23:49

Yep, that exactly right, although it's always best to explicitly say what the point would be for a specific n.

And for d_2, you should be able to find an expression for the area under the curve f_n, since it is only a triangle. Now, show that the area ->0 as n->inf.

Arcade Fire · 8 Feb 2007 at 23:51

Psiko said:
Yep, that exactly right, although it's always best to explicitly say what the point would be for a specific n.

And for d_2, you should be able to find an expression for the area under the curve f_n, since it is only a triangle. Now, show that the area ->0 as n->inf.

Sorry, are we looking at the same question here? The area under the curve |f_n(x)-f(x)| where f is the zero function is clearly *not* a triangle.

Psiko · 8 Feb 2007 at 23:53

Arcade Fire said:
Sorry, are we looking at the same question here? The area under the curve |f_n(x)-f(x)| where f is the zero function is clearly *not* a triangle.

Lol. Sorry, ignore that. I meant rectangle.

titaniumx3 · 9 Feb 2007 at 00:09

Yeah, its more like a rectangle that gets progressively 'thinner'.

Thanks again to all for the help, I've actually begun to understand some of the lecture notes better now.

Arcade Fire · 9 Feb 2007 at 00:17

As with most analysis, the key to understanding metric spaces is to keep the definitions close at hand at all times, until you're so comfortable with them that they're second nature. When you're given a new definition it's always a good idea to come up with a few examples yourself and see how they figure in the new framework. That should give you a 'feel' for how the definition works, and how it slots in with what you already know. For example, you should be able to see, using a few examples, that a metric is just like a measure of distance, only more general.

To actually solve problems you need to use your intuition somewhat to gauge how best to approach the problem, but a full solution can only come from applying the definitions systematically, so that's why you need to keep them in mind. I found when answering analysis questions that even when I had no idea what was going on, I could still make good progress (and sometimes even complete the question) just by writing down any relevant definitions and theorems and just doing what seemed natural from there.

Good luck!

titaniumx3 · 9 Feb 2007 at 01:02

That is very true and something I always try to do when approaching new definitions and theorems; just writing down a few basic examples of the theorem in practice helps a lot and as you say once you get a 'feel' for it the problems pretty much start to roll out themselves. The main issue I have though is that quite frequently it just takes rediculously long to actually comprehend the theorems/definitions and how exactly to apply them; don't even get me started on some of the proofs! :eek: