Home work to be handed in April 15
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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.
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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.
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Characterise the equality cases in Young's, Holder and the
triangle inequality (for the p-norm on R^n).
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Prove that every norm on R^n is continuous with respect to
(your favorite) standard norm on R^n.
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Redo 1 and 2 above for the case that K is a subspace with
equalities for inner products.
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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.
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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).
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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.