104 HW2 .pdf - Homework#2 Math 104A Pedro Aristizabal Posada 1 Let V be a vector space Prove that a norm ∥ � ∥ on V defines a continuous function

104 HW2 .pdf - Homework#2 Math 104A Pedro Aristizabal...

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Homework #2 Math 104A Pedro Aristizabal Posada 1.Let V be a vector space. Prove that a norm ∥ · ∥ on V defines a continuous function ∥·∥:V [0,). We start by defining a function f : V [0,inf) as f(x) = ||x|| for all x in V. Now that our function is defined, we are asked to show that ||.|| on V can define our continuous function f(x). We can start by using a triangle inequality for x,y to show: .
Homework #2 Math 104A Pedro Aristizabal Posada ||x|| 2 = sqrt(|x 1 | 2 + |x 2 | 2 ) ||x|| inf = max{ |x 1 |, |x 2 | }
Homework #2 Math 104A Pedro Aristizabal Posada

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