Due August 14
Problem 1
The function dT(x,y)=|x1−y1|+…+|xn−yn| 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||2≤c2||x||1
for all x∈X.
- The norms ||⋅||E,||⋅||M and ||⋅||T on \Rn are equivalent.
- 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).
- 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.
- 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 f∈C([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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