Cauchy-Schwartz inequality question

[nikhil714]

Let M ⊂ H be a closed linear subspace that is not reduced to {0}. Let f ∈ H,f /∈ M⊥.

Prove that m = inf (f, u) is uniquely achieved.
u∈M
|u|=1



I have approached it with the cosine definition for inner product but that only works with Euclidean spaces. I want to know how to apply this using the Cauchy-Schwartz inequality so it works for Banach Spaces. Can anyone help? I have placed my work below.




The Hilbert Projection Theorem says that there exists a unique object in M that minimizes the distance to f. It also happens to be the projection.

So let f = p + q, where p is in M, and q is in Mperp. (We use the projection theorem on M and Mperp s.t. p is the projection onto M, and q is the projection onto Mperp)

(u,f) = (u, p+q) = (u,p) + (u,q)

Now (u,q) = 0 because u in M, q is in Mperp so (u,f) = (u,p).


(u,p) = ||u|| ||p|| cos(theta) = ||p|| cos(theta)

this is uniquely minimized when cos(theta)=-1, so u is on the opposite side in M, but with a norm of 1 (it is one away from the origin and thus Mperp)

u = -p/(||p||)

m = (u,p) = (-p/||p||,p) = -(p,p)/||p|| = -||p||^2 / ||p|| = -||p||

[blue89]

Hi ,

 

I am unsure but again I think you migh use one of the proof of cauchy schwars inequality to take help.

 

Remember we are beginning to prove it :

 

0 < II u - λ.v II → (to go on..)

R: https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality (its proof might provide you a method

and this also helps us to show minkowski inequality which we use it to create a useful expression for normic spaces or hilberd (remember please l_p was a hilberd space only if only when p = 2 )

 

minkowski inequality is :

 

( ∑ (ai + bi ) p )1/p ≤ ( ∑ ai p )1/p + ( ∑ bi p )1/p

there, i= 1,2,..,n p >1

 

furhermore , the Hilbert Projection theorem is correct there why as you wrote M ⊂ H and M is closed . M is complete because H is Hilberd this means any cauchy sequence which its elements are chosen from M , will be convergent and its limit will be elemnt of M

 

then , of course Hilberd projection is provided (correct) there (convexity is also correct )

 

But I did see some invisible statement or these are not enough to make interpretations

 

 

 

 

I have approached it with the cosine definition for inner product but that only works with Euclidean spaces. (???) I want to know how to apply this using the Cauchy-Schwartz inequality so it works for Banach Spaces

 

look , some euclidean spaces are also both banach space or at the same time HILBERD

 

e.g : R³is banach space (Euclidean (✔))

l₂ is hilberd.

 

R: limited dimensional Euclidean spaces are Hilberd https://en.wikipedia.org/wiki/Hilbert_space

R: https://en.wikipedia.org/wiki/Banach_space

 

but if you meant rieamnnian manifolds or .. (I do not have enough information (Geometry))

 

 

and what have you implied for its actual meaning via using this notation

 

 

 

Mperp

 

is this just in the meaning marginal elements of M ??

Edited September 2, 2016 by blue89