Home work to be handed in April 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Projection of x on a closed convex subset K of a real Hilbert space by finding the nearest point y to x in K. 1. Formulate the minimizing property in terms of inequalities for inner products involving x,y, and an arbitrary point z in K. 2. Prove that this formulation is equivalent to y being nearest to x in K. 3. Show that the map P that takes x to y, x and y as above, is contractive. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx p-norms in the plane. Let x be in R^2. Prove that the p-norm of x converges to the inifinity norm of x as p goes to infinity. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Characterise the equality cases in Young's, Holder and the triangle inequality (for the p-norm on R^n). xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Prove that every norm on R^n is continuous with respect to (your favorite) standard norm on R^n. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Redo 1 and 2 above for the case that K is a subspace with equalities for inner products. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx If L is a finite-dimensional subspace of a normed space X, L not equal to X, show that there exists a y in L such that ||y||=1 and ||x-y|| is larger or equal to 1 for all x in L. You may use that for every z not in L there exists a point in L which is nearest to z. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Prove that the functions x_n:[0,1] -> R defined by x_n(t)=t^n have no convergent subsequence (with respect to the maximumnorm). xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Let X be a vector space and F: X-> R linear, F not the zero-function. Prove that the kernel of F, N(F)={x in X: F(x)=0} has co-dimension one (define what this means). Prove that if also G: X -> R is linear, with G(x) =0 for all x in N(F), that G=aF for some a in R.