Cauchy-Schwarz Inequality for Summations

[steelphantom]

Homework Statement

Prove that (\sum_{j=1}^na_jb_j)² <= (\sum_{j=1}^nja_j²)(\sum_{j=1}^n(1/j)b_j²)

Homework Equations

Cauchy-Schwarz Inequality: |<u, v>| <= ||u||*||v||

The Attempt at a Solution

If I let u = \sum_{j=1}^nja_j² and v = \sum_{j=1}^n(1/j)b_j², then I have ||u|| = sqrt(<u, u>) and ||v|| = sqrt(<v, v>). Not really sure where to go from here. Any ideas? Thanks!

 

[Vid]

The other form is more helpful in this problem:
<u,v>^2 = <u,u><v,v>

Now, you just have to choose u and v to get the required identity. You know that <u,v> needs to equal to Sum[a_j*b_j]. You also know that the j-th term in the series on the right side is equal to the j-th component of u or v squared.

Can you finish it from here?

 

[steelphantom]

The other form is more helpful in this problem:
<u,v>^2 = <u,u><v,v>

Now, you just have to choose u and v to get the required identity. You know that <u,v> needs to equal to Sum[a_j*b_j]. You also know that the j-th term in the series on the right side is equal to the j-th component of u or v squared.

Can you finish it from here?

First of all, thanks for the help!

Don't you mean <u,v>² <= <u,u><v,v> ? Also, I'm not sure what you mean by your last sentence. "The series on the right side" is referring to which series? The product of the two series? Anyway, how do I know that? Sorry for these possibly stupid questions, and thanks again for your help.

 

[Vid]

Yea, that should be less than or equal to.

I meant that one series was for u and the other for v. With the standard dot product <u,u> = Sum[(u_j)^2].

So Sum[u_j^2] = Sum[j*a_j^2]
and Sum[v_j^2] = Sum[(1/j)*b_j^2]

What does this tell us about what u and v must be equal to?

 

[steelphantom]

Yea, that should be less than or equal to.

I meant that one series was for u and the other for v. With the standard dot product <u,u> = Sum[(u_j)^2].

So Sum[u_j^2] = Sum[j*a_j^2]
and Sum[v_j^2] = Sum[(1/j)*b_j^2]

What does this tell us about what u and v must be equal to?

I don't think I'm seeing it. u_j = sqrt(j)*a_j and u = Sum[u_j] = Sum[sqrt(j)*a_j] ?
v_j = (1/sqrt(j))*b_j and v = Sum[v_j] = Sum[(1/sqrt(j))*b_j] ?

Is this correct? I don't really see how this helps.

 

[ircdan]

the cs inequality says <p, q> <= |p||q|, here | | denotes the norm

so (<p, q>)^2 <= |p|^2|q|^2


Set p = (a_1/sqrt(1),..., a_n/sqrt(n)), q= (sqrt(1)*b_1, 2*b_2,..., sqrt(n)*b_n)


so your sum on the left = (<p, q>)^2 <= |p|^2|q|^2 = the stuff you have on the right

just verify this

 

[steelphantom]

Ok, just to make sure I got it:

Cauchy Schwarz says |<u, v>| <= ||u||*||v||. This implies <u, v>² <= <u, u><u, v>. Let <u, u> = \sumu_j² = \sumja_j² and let <v, v> = \sumv_j² = \sum(1/j)b_j².

Then we have u = \sumu_j = \sumsqrt(j)*a_j and v = \sumv_j = \sum(1/sqrt(j))*b_j.

Then we have <u, v> = \sumu_jv_j = ... = \suma_jb_j. Then square both sides to get <u, v>² = (\suma_jb_j)².

Then after plugging in, this completes the proof. Thanks again for all your help guys!