Written assignment 2 (3 points): Prove MF thm.12.1 (Norms define metric spaces): Let (V, I. D be a normed vector space.

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Written Assignment 2 3 Points Prove Mf Thm 12 1 Norms Define Metric Spaces Let V I D Be A Normed Vector Space 1 (132.33 KiB) Viewed 27 times

Written assignment 2 (3 points): Prove MF thm.12.1 (Norms define metric spaces): Let (V, I. D be a normed vector space. Then the function d):V XV Ro (x, y) Hd (2,y) := \y – 2|| defines a metric space (V, 01.11). This assignment is worth three points: One point each for pos.definite, symmetry, triangle inequality! Hint: You will have to show for each one of (12.1a), (12.1b), (12.1c) how it follows from def. 11.15: Which one of (11.31a), (11.31b), (11.31c) do you use at which spot? Careful with symmetry: What is the reason that ||a – b1 = || b – a||? This assignment is worth three points, and you will have to earn them! The following exemplifies the level of detail I expect you to provide. To prove, e.g., that d(,) satisfies the triangle inequality (12.1c) of a metric you will have to write something along the following lines: Triangle inequality. NTS://(x,z) Ş0(2,y) + (y, z) for all x, y, 2 EX. Proof: dyr,z) = ||z – (definition of the metric d. c. S... (.....) (2,y) + d(y,z) (.....) (There may be fewer or more steps in your proof. The above only serves as an illustration!)