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Tarea 1, Análisis real

Due August 14


Problem 1


The function dT(x,y)=|x1y1|++|xnyn| defined for x,y\Rn is a metric on \Rn.

Problem 2


Two norms ||||1 and ||||2 on a vector space are equivalent if there exist constants c1,c2 such that

c1||x||1||x||2c2||x||1


for all xX.

  1. The norms ||||E,||||M and ||||T on \Rn are equivalent.

  2. If ||||1 and ||||2 are equivalent and Bir(x) is the ball of radius r with center x with respect to the metric induced by ||||i, then, for each \e>0, there exist δ1,δ2>0 such that B1δ1(x)B2\e(x) and B2δ2(x)B1\e(x).

  3. Let ||||1 and ||||2 two norms on X, and suppose there exist δ,\e>0 such that B1δ(0)B21(0) and B2\e(0)B11(0). Then ||||1 and ||||2 are equivalent.

  4. Equivalent norms induce the same topology; i. e., U is open in (X,||||1) if and only if it is open in (X,||||2).


Problem 3


Let ||||1 and ||||2 be norms on \Rn induced by inner products. Then they are equivalent.

Problem 4


The uniform, L1 and L2 norms on C([0,1]) satisfy

||f||1||f||2||f||u


for all fC([0,1]). However, not two of them are equivalent.

Problem 5


Let ˉB\e(x) be the closed ball with center x and radius \e>0, and ¯B\e(x) the closure of the open ball B\e(x). Then ¯B\e(x)ˉB\e(x). Give an example where ˉB\e(x).

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