Real analysis metric/norm question

label Mathematics
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Apr 27th, 2015

It is a known result that if V is a vector space and m(x,y) is a metric on V, then m induces a norm on V defined by ||x|| = m(x,0) where x, y are elements of V only if both

(1) m(x,y) = m(x+a,y+a)
and
2) m(alpha*x,alpha*y) = |alpha|*m(x,y).

Since the problem doesn't quality the vector space, I assume that V = R (the real numbers).

A) e(x,y) does induce a norm on R because both (1) and (2) are satisfied:

e(x,y) = sqrt((x-y)^2) = |x-y|
e(x+a,y+a) = sqrt(((x+a)-(y+a))^2) = sqrt((x-y)^2) = |x-y| = e(x,y)

e(alpha*x,alpha*y) = sqrt((alpha*x-alpha*y)^2) = sqrt(alpha^2 *(x-y)^2) = |alpha| * |x-y| = |alpha|*e(x,y).

B) d(x,y) does not make a norm on R because it fails property (1)
C) s(x,y) does not make a norm on R because it fails property (2)

 


Apr 27th, 2015

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