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Unformatted text preview: Solutions for HWK 4 Based on the HWKs by my former student JINGYUAN LIU Problem A Let  x  be a norm in R d . Prove that ρ ( x , y ) :=  x y  is a metric in R d , i.e. (see p. 128) a) ρ ( x , y ) ≥ , ∀ x , y ∈ R d ; ” = ” holds iff x = y . b) ρ ( y , x ) = ρ ( x , y ) (”symmetry”). c) ρ ( x , y ) + ρ ( y , z ) ≥ ρ ( x , z ) ∀ x , y , z ∈ R d (”the triangle inequality”). Proof . Recall the properties of the norm: (1)  x  ≥ ∀ x ∈ R d ; (2)  x  = 0 iff x = ; (3)  α x  =  α   x  ∀ α ∈ R , x ∈ R d ; (4) ∀ x , y ∈ R d the ”triangle inequality” holds:  x + y  ≤  x  +  y  . Put α = 1 in (3). We have : 3 )   x  =  x  . By definition, ρ ( x ,y ) =  x y  . Let us verify that ρ ( x , y ) possesses the properties a)c) of a metric. (a) Since  x y  is a norm, (1) tells us  x y  ≥ 0 hence ρ ( x , y ) ≥ , ∀ x , y ∈ R d ....
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This note was uploaded on 10/13/2010 for the course MATH 403 at Penn State.
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