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Question: engineering analytical techniques vector spaces hw 190125 department of electrical...

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ENGINEERING ANALYTICAL TECHNIQUES Vector Spaces HW 19_0125 Department of Electrical and Computer Engineering University of Miami Question Points Question 20 Question 2 20 Question 320 Question 4 20 Question 5 20 I. QUESTION 1 Exercise 2.1-2 of Moon and Stirling, 2000]: Let X be an arbitrary set. Show that the function defined by 0, for x=y: d(x,y) = 1, for x#y is a metric II. QUESTION 2 Exercise 2.1-5 of [Moon and Stirling, 2000: Let (X,d) be a metric space. Show that d(x,y) is a metric on X. What significant feature does this metric possess?III. QUESTION 3 Exercise 2.1-7 of [Moon and Stirling, 2000]: In defining the metric of the sequence space (0, 0) as d(, y)sup(n)-v(n), sup is used instead of max. To see the necessity of this definition, define the sequences {x(n)) and ty(n)) by n+ 1 Show that doo(z, y) > |x(n)-y(n). Yn 21 IV. QUESTION 4 Exercise 2.1-12 of [Moon and Stirling, 2000: Let (a) Draw the set B (b) Determine the boundary of B (c) Determine the interior of B V. QUESTION 5 Exercise 2.1-24 of [Moon and Stirling, 2000]: The fact that a sequence is Cauchy depends upon the metric employed. Let /n (t) be the sequence of functions for t <-1/n; (t)-nt/2+1/2, for-1/n sts 1/n; for 1/n <t, in the metric space (C[a, b,, where doo (f,g) sup f (t) -g(t) Show that = . for rn > n 2 2m Hence, conclude that in this metric space, fn is not a Cauchy sequence.

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