Written assignment 1: (3 points!) Prove MF prop.11.13 (Properties of the sup norm): h ||hl|oo = sup{]h(x): EX} defines a

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Written Assignment 1 3 Points Prove Mf Prop 11 13 Properties Of The Sup Norm H Hl Oo Sup H X Ex Defines A 1 (134.46 KiB) Viewed 27 times

Written assignment 1: (3 points!) Prove MF prop.11.13 (Properties of the sup norm): h ||hl|oo = sup{]h(x): EX} defines a norm on B(X, R) This assignment is worth three points: One point each for pos.definite, absolutely homogeneous, triangle inequality! Hint: Go for a treasure hunt in ch.9.2 (Minima, Maxima, Infima and Suprema), and look at the properties of sup(A) (A CR) to prove absolute homogeneity and the triangle inequality. 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 that || - lloc satisfies the triangle inequality (11.27c) of a norm you will have to write something along the following lines: Triangle inequality. NTS: 1 $ +9||. ||$||00 + ||$||- for all f, g EB(X, R). Proof: Ilft glloo sup{\f(x)]+\g(x): 1 EX} (definition of I. ||.) C. . || f ||- + ||g|| (.....) (There may be fewer or more steps in your proof. The above only serves as an illustration! No need to justify properties of the absolute value la of a real number a, but you will need to justify why sup{laf (3): ex} = |a|sup{If(): TEX), and why sup{\S () +9(2): EX} sup{[/(w): * EX} + sup{9(2): EX}. An aside: DO NOT write ||| (2) ||e when you deal with the real number $() (and you probably mean the absolute value f(x)). lloc is defined for functions f, NOT for numbers f(x)!