Homework help, Banach spaces...

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haffi88112
Posts: 3
Joined: Sat Feb 28, 2009 4:54 pm UTC

Homework help, Banach spaces...

This has been giving me a headache...

Let C([a;b]) be the set of all continuous real functions on the closed interval [a;b]. Then C([a;b]) is an algebra over the real numbers and further more a vector space over the reals.

Now we define a norm on this space so that

||f||_2 = sqrt("Integrate from a to b"(|f(t)|^2)dt) (This is a Riemann integral, not Lebesque afaik)

Now I'm supposed to show that with this norm this is not a Banach Space...

I've been making sequences of functions all day trying to find a cauchy sequence that converges outside of C([a;b]) but I haven't succeeded. Maybe I'm trying the wrong method to solve this but I can't think of another way.

Help?

Torn Apart By Dingos
Posts: 817
Joined: Thu Aug 03, 2006 2:27 am UTC

Re: Homework help, Banach spaces...

Try to think of a sequence of continuous functions such that f_n(x) tends to 0 for x in [0,1) and f_n(x) tends to 1 for x in [1,2].

haffi88112
Posts: 3
Joined: Sat Feb 28, 2009 4:54 pm UTC

Re: Homework help, Banach spaces...

f_n(x) = (x^(2n))/(1+x^(2n)) works but I can't see how that helps.

hnooch
Posts: 128
Joined: Mon Nov 26, 2007 6:55 pm UTC

Re: Homework help, Banach spaces...

Is that a Cauchy sequence? If so, does it converge (under your norm) to a function which is continuous on [0,2]? If it had to converge to some function, what would that function be?

haffi88112
Posts: 3
Joined: Sat Feb 28, 2009 4:54 pm UTC

Re: Homework help, Banach spaces...

Ok, I think I can finish this now.

I just need to show that it's a Cauchy sequence because the limit is not continuous.

Torn Apart By Dingos
Posts: 817
Joined: Thu Aug 03, 2006 2:27 am UTC

Re: Homework help, Banach spaces...

Well, actually, there are lots of limits (in the space of all square integrable functions). You would need to show that they are all discontinuous.

ThomasS
Posts: 585
Joined: Wed Dec 12, 2007 7:46 pm UTC

Re: Homework help, Banach spaces...

Torn Apart By Dingos wrote:Well, actually, there are lots of limits (in the space of all square integrable functions). You would need to show that they are all discontinuous.

Good catch. To show that a normed vector space fails to be a Banach space you just need to show that it isn't complete. Which is to say that there is one Cauchy sequence which does not have a limit. So strictly speaking the statement that it's limit is not continuous is non-sensical. On the other hand, it shouldn't be hard to prove that if there were a limit it would be equal to 0 almost everywhere on a certain subinterval and so on.