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. ***************************************************** 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. ******************************************************* 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. ******************************************************* 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. ******************************************************* Show that l^(1) is isometric to the dual of (c_0), see http://www.few.vu.nl/~jhulshof/FA2008/week4/DSC01936.JPG.