Hilbert Spaces Help: Let M ⊂ H be a closed linear subspace that is not reduced to {0}

[nikhil714]

Hilbert Spaces Help: Let M ⊂ H be a closed linear subspace that is not reduced to {0}

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

 

[nikhil714]

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

So this is what I have been working with based off of the definitions and examples I have seen.

I can decompose objects in a Hilbert space into the part in M and the part in Mperp

The Hilbert Projection Theorem says that there exists a unique object in M that minimizes the distance to f, which is the projection.

So let f = p + q, where p is in M, and q is in Mperp. (I think I can use the projection theorem on M and Mperp such that 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).

As you can see, inner producting with something from a subspace only takes into account the part in that subspace.

Now, I'm not too sure what to do with this:

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



Still quite lost with parts 2 and 3 of the problem.