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ECEN601_hw3_pfister

ECEN601_hw3_pfister - ECEN 601 Assignment 3 Problems 1(TOP...

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Unformatted text preview: ECEN 601: Assignment 3 Problems: 1. (TOP: 2.7.2) Suppose that f : X → Y is continuous. If x is a limit point of the subset A of X , is it necessarily true that f ( x ) is a limit point of f ( A )? 2. (TOP: 2.9.8) Show that the Euclidean distance d on R n is a metric, as follows: If x , y ∈ R n and c ∈ R , define x + y = ( x 1 + y 1 ,...,x n + y n ) c x = ( cx 1 ,...,cx n ) , x · y = x 1 y 1 + ··· + x n y n bardbl x bardbl = ( x · x ) 1 / 2 . (a) Show that x · ( y + z ) = ( x · y ) + ( x · z ). (b) Show that | x · y | ≤ bardbl x bardblbardbl y bardbl . [Hint: If x , y negationslash = 0, let a = 1 / bardbl x bardbl and b = 1 / bardbl y bardbl , and use the fact that bardbl a x ± b y bardbl ≥ 0.] (c) Show that bardbl x + y bardbl ≤ bardbl x bardbl + bardbl y bardbl . [Hint: Compute ( x + y ) · ( x + y ) and apply previous result.] (d) Verify that the Euclidean distance d ( x , y ) is a metric....
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ECEN601_hw3_pfister - ECEN 601 Assignment 3 Problems 1(TOP...

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