<uk,v> -> <u,k> for all v. prove uk goes to u.

[cap.r]

<uk,v> --> <u,k> for all v. prove uk goes to u.

Homework Statement

a)supposed for all v in Rⁿ <u_k,v>--> <u,v> prove {u_k}-->u
b)give an example of a sequence {u_k} st. <u_k,v> for some vector v. but u_k does not converge to u.

Homework Equations

def of convergence.

The Attempt at a Solution

a)since this works for all v. we can let v₁={1,0,0,...} then u₁ --> u. now let v₂={0,1,0,0,...} then u₂ --> u.
so we keep doing this and each term of u_k converges to a term in u.
so by point wise convergence we have convergence.

b) I don't know how to do this part. for a) i think i got it right but i can't come up with an example.

 

[aPhilosopher]

your proof looks good. For the example, note that if u_k converges to u then u_k + w converges to u + w. Can you think of a way to choose w so that <u_k + w, v> converges to <u, v>?

 

[LCKurtz]

You have the right idea for part 1 but you aren't writing it very well. Remember that each u_k is a vector. So it doesn't make sense to say u₁ --> u. u₁ is a fixed vector. It might help you to adopt a notation where the components of, for example u₁ are u₁(1), u₁(2),...u₁(n), and try making the argument you are thinking of.

Once you get the notation down you may see how to do the second part too.