HW202 - Analysis 2 Homework 02 1. Let H be a real Hilbert...

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Analysis 2 Homework 02 1. Let H be a real Hilbert space on which φ is a bounded linear functional; define f : H R : x 7→ k x k 2 + φ ( x ) . Prove that each (nonempty) closed convex K H contains a unique point at which the restriction f | K is minimized. Solution 1. Recall from class the theorem that each closed convex set K in H contains a unique point of minimal norm. We could adopt its method of proof . Note first that if x,y K then (parallelogram) 1 2 k x - y k 2 = k x k 2 + k y k 2 - 2 k 1 2 ( x + y ) k 2 whence rewriting squared norms in terms of f and φ (which cancels by lin- earity) yields 1 2 k x - y k 2 = f ( x ) + f ( y ) - 2 f ( 1 2 ( x + y )) . Now, let δ = inf { f ( z ) : z K } and for each n choose z n K such that ( δ 6 ) f ( z n ) < δ +1 /n . As 1 2 ( x + y ) K it follows from above that if p,q > n then 1 2 k z p - z q k 2 < ( δ + 1 /p ) + ( δ + 1 /q ) - 2 δ < 2 /n thus ( z n : n > 1) is Cauchy and so converges .
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HW202 - Analysis 2 Homework 02 1. Let H be a real Hilbert...

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