exercise1 - MA3056 — Exercise Sheet 1: Metric Spaces...

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Unformatted text preview: MA3056 — Exercise Sheet 1: Metric Spaces 15/1/04 1. (a) * Prove that the following inequality holds for any n ≥ 1 and a i ,b i ∈ [0 , ∞ ): ∑ n i =1 a i b i ≤ (∑ n i =1 a 2 i ) 1 / 2 (∑ n i =1 b 2 i ) 1 / 2 ( ‡ ) [Hint: consider the quadratic polynomial p ( x ) = ∑ i ( a i- xb i ) 2 .] Hence show that the map d 2 defined as follows is a metric on C n d 2 ( ( z 1 ,...,z n ) , ( w 1 ,...,w n ) ) = (∑ n i =1 | z i- w i | 2 ) 1 / 2 (b) Prove that if f : [0 , 1] → [0 , ∞ ) is a continuous function then Z 1 f ( t ) dt = 0 ⇔ f ( t ) = 0 ∀ t ∈ [0 , 1] . Hence show that the map d 1 defined as follows is a metric on C [0 , 1] d 1 ( f,g ) = Z 1 | f ( t )- g ( t ) | dt [To show that the function d 2 ( f,g ) = R 1 | f ( t )- g ( t ) | dt 1 / 2 is a metric on C [0 , 1] requires an analogue of ( ‡ ).] 2. Find three different metrics on N , no two of which are multiples of each other....
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exercise1 - MA3056 — Exercise Sheet 1: Metric Spaces...

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