Continue to Site

Welcome to EDAboard.com

Welcome to our site! EDAboard.com is an international Electronics Discussion Forum focused on EDA software, circuits, schematics, books, theory, papers, asic, pld, 8051, DSP, Network, RF, Analog Design, PCB, Service Manuals... and a whole lot more! To participate you need to register. Registration is free. Click here to register now.

Register Log in

Help me prove two statements for Introduction to Hilbert Spaces course

Status
Not open for further replies.
S

silveredition

Newbie level 6
Joined
Sep 21, 2006
Messages
11
Helped
2
Reputation
4
Reaction score
0
Trophy points
1,281
Activity points
1,372
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?
 

Re: Proof Question

Seems like you Haven't taken Calculus in awhile :D .

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.
 

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.
 

Re: Proof Question

silveredition said:
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.
Click to expand...




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.
 

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.
 

Re: Proof Question

hadi_hdk said:
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.
Click to expand...

g(x) is continuous
 

Status
Not open for further replies.

Similar threads

Part and Inventory Search

Welcome to EDABoard.com

Sponsor

Back
Top