Let X and Y be normed spaces, A: X -> Y a continuous linear map.
The (operator) norm of A, ||A||, is defined as the infimum of all M>0
such that ||Ax|| is less or equal than M ||x|| for all x in X.
(i) Show that ||Ax|| is less or equal than ||A|| ||x|| for all x in X.
(see also http://www.few.vu.nl/~jhulshof/FA2008/week3/DSC01871.JPG for the
case that X=H and Y=R)
(ii) Show that the operator norm defines a norm on B(X,Y), the set of all
continuous linear maps from X to Y.
(iii) Prove that B(X,Y) is complete if Y is.
See http://www.few.vu.nl/~jhulshof/FA2008/week3/DSC01880.JPG.
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Let A: R^n -> R^n be a symmetric linear map.
Prove that x -> |Ax|| and x -> | Ax . x | have
the same maximum on the unit sphere {x in R^n: ||x||=1 }
Hint: call the maxima M1 and M2. M2 smaller or equal M1
should be easy. The other way around requires a trick,
you should consider
A(x+y) . (x+y) - A(x-y) . (x-y) = ....
The left hand side is bounded by ....
In the right hand side you choose y=...
Prove the same statement for A : H -> H symmetric, bounded, linear,
when H is a real Hilbert space, with maximum replaced by supremum.
Prove that any x in H which maximizes | Ax . x | on the unit ball
is an eigenvector (so sup=max implies that A has an eigenvector)
with a real eigenvalue. Hint: the proof is the same for H as for R^2.
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Let H be a Hilbert space.
(i) If L is a finite-dimensional subspace of H, spanned
by an ortonormal set phi_1,....,phi_n, write down and prove a
formula for the orthogornal projection Px on L of x in H.
(ii) If L is a subspace of H, define the orthoplement of L
to be the set of all x in H for which (x,y)=0 for all y in L.
Show that the orthoplement of L is a closed linear subspace of H.
(iii) Show that L is contained in the orthoplement of the orthoplement of L.
(iv) Show that the closure of L is contained in the orthoplement of the orthoplement of L.
(v) Show that the closure of L is equal to the orthoplement of the orthoplement of L.
Hint: show that every x can be written uniquely as y+z with y in the
closure and z in orthoplement of L.
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Prove that l^(p) is a Banach space for all p larger or equal than 1,
see http://www.few.vu.nl/~jhulshof/FA2008/week4/DSC01926.JPG.
Prove that l^(p) is separable if p is finite,
see http://www.few.vu.nl/~jhulshof/FA2008/week4/DSC01927.JPG,
see http://www.few.vu.nl/~jhulshof/FA2008/week4/DSC01928.JPG.
Prove that (c) and (c)_0 are separable Banach spaces with respect to the supremumnorm,
see http://www.few.vu.nl/~jhulshof/FA2008/week4/DSC01929.JPG.
Prove that the parallellogram law holds for the p-norm in l^(p) if and only if p=2.
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Show that l^(1) is isometric to the dual of (c_0),
see http://www.few.vu.nl/~jhulshof/FA2008/week4/DSC01936.JPG.