Homework help, Banach spaces...

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haffi88112
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Homework help, Banach spaces...

Postby haffi88112 » Sat Feb 28, 2009 5:48 pm UTC

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?

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Torn Apart By Dingos
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Re: Homework help, Banach spaces...

Postby Torn Apart By Dingos » Sat Feb 28, 2009 6:47 pm UTC

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
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Re: Homework help, Banach spaces...

Postby haffi88112 » Sat Feb 28, 2009 6:59 pm UTC

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

hnooch
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Re: Homework help, Banach spaces...

Postby hnooch » Sat Feb 28, 2009 7:46 pm UTC

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
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Re: Homework help, Banach spaces...

Postby haffi88112 » Sat Feb 28, 2009 8:12 pm UTC

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.

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Torn Apart By Dingos
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Re: Homework help, Banach spaces...

Postby Torn Apart By Dingos » Sat Feb 28, 2009 9:49 pm UTC

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
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Re: Homework help, Banach spaces...

Postby ThomasS » Sun Mar 01, 2009 1:46 am UTC

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.


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