Help me prove two statements for Introduction to Hilbert Spaces course

[silveredition]

Haven't taken Calculus in awhile, and have to prove two statements for my Introduction to Hilbert Spaces course. I was hoping someone may be able to enlighten me. Essentially, I have worked these two problems down to the point where I have to prove the following:

1) g(x) non-negative on [0 2], and continuous
show that if the mean of g(x) = 0 over this interval, then g(x) must = 0

2) Let { fn } be a sequence of functions with the following property:

fn converges
for each fn(x), lim of fn(x) as |x|->infinity = 0

If f(x) = lim of fn(x) as n->infinity (the convergence of the sequence), then prove that
lim of f(x) as |x|->infinity = 0 as well.

Any ideas?

 

[mayyan]

Re: Proof Question

Seems like you Haven't taken Calculus in awhile .

i give you two non regorous proves.

1) the mean of a non negetive function over an interval greater then zero must be >= 0 ( its the intergal divided by the interval length). => f(x) must be equal to zero.

2) for each Fn(x) we have Fn(x) -> 0 (as x-> infinity). this means:
F1(infinity) -> 0
F2(infinity) -> 0
F3(infinity) -> 0
.
.
.
Fn(infinity) -> 0

since Fn(x) -> f(x) (as n-> infinity) then f(infinity) must be f(infinity)->0 otherwise the original stament [Fn(x) -> f(x) (as n-> infinity) ] isnt true.

one more thing. try drawing it. it will give meaning to what i said above.

 

[silveredition]

Re: Proof Question

As far as 1) goes, I just wasn't sure if there was some theorem that concretely states that if the mean of the function is 0 and it's non-negative that it has to be equal to 0. As far as 2) goes, it's easy to see that each Fn(x)->0 as |x|->inf by the definition, but I don't need to show that the fn(x) do that, since we already know that, I need to prove that whatever function they end up converging to does.

 

[mayyan]

Re: Proof Question

As far as 1) goes, I just wasn't sure if there was some theorem that concretely states that if the mean of the function is 0 and it's non-negative that it has to be equal to 0. As far as 2) goes, it's easy to see that each Fn(x)->0 as |x|->inf by the definition, but I don't need to show that the fn(x) do that, since we already know that, I need to prove that whatever function they end up converging to does.

thats what i have proven. since all Fn(infinity)->0 and Fn(x)->f(x) then f(infinity)->0. draw it and you will see it.

 

[hadi_hdk]

Re: Proof Question

1. you cant proof this. if you suppose that g(x) in some finite point equal to 1 and in all other point equall zero then mean of g(x) are zero but g(x)≠0.

 

[mayyan]

Re: Proof Question

1. you cant proof this. if you suppose that g(x) in some finite point equal to 1 and in all other point equall zero then mean of g(x) are zero but g(x)≠0.

g(x) is continuous