math403 hw4 - Solutions for HWK 4 Based on the HWKs by my...

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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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math403 hw4 - Solutions for HWK 4 Based on the HWKs by my...

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