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solutionshw2-201141 - Solutions for HW 2 Problem 25:8 Let...

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Solutions for HW 2 Problem 25:8. Let x n x , y n y in the normed space V and α n α , β n β in the scalars. Then we have k α n x n - αx k = k α n x n - α n x + α n x - αx k ≤ | α n |k x n - x k + | α n - α |k x k . Now ( α n ) is bounded, so | α n |k x n - x k → 0, and similarly | α n - α |k x k → 0. Hence we have k α n x n - αx k → 0, similarly we have k β n y n - βy k → 0. The first part of the problem follows now from k ( αx + βy ) - ( α n x n + β n y n ) ≤ k α n x n - αx k + k β n y n - βy k . In case of an inner product space we use the same “add and subtract trick”. In this case we use that if ( x n ) converges than ( k x n k ) bounded, which follows from k x n k ≤ || x n - x k + k x k . Now we have | x n · y n - x · y | ≤ | x n · y n - x · y n + x · y n - x · y | ≤ | ( x n - x ) · y n | + | x · ( y n - y ) | ≤ k x n - x kk y n k + k x kk y n - y k → 0 . Problem 30: 2. Let E = { x : d ( x, y ) r } be a closed ball. Let x be a limit point of E . Then there exists a sequence ( x n ) in E such that x n x . Now let > 0. Then there exists N such that d ( x, x n ) < for all
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