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: ||x|| = ||x y + y|| ≤ ||x - y|| + ||y|| ||x|| - ||y|| ≤ ||x -y|| (given property) If we re-write this as | ||x||- y|| | ≤ ||x – y||, we can make the argument | f(x) -f(y) | = | ||x|| - ||y|| | ≤ ||x -y|| After stating this, we know for each ε > 0, there must be a value d such that |f(x) f(y)| < ε whenever ||x y|| < d. By showing this, we know f is continuous at all points in V, showing that if ||.|| is continuous, so is f. 2. Let V = R 2 . Sketch the unit ball for the norms ∥·∥ 1 , ∥·∥ 2 ,and ∥·∥ . We know V = R^2 , so : ||x|| 1 = |x 1 | + |x 2 |
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Homework #2 Math 104A Pedro Aristizabal Posada ||x|| 2 = sqrt(|x 1 | 2 + |x 2 | 2 ) ||x|| inf = max{ |x 1 |, |x 2 | }
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Homework #2 Math 104A Pedro Aristizabal Posada
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