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)and2) 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)
Content will be erased after question is completed.
Enter the email address associated with your account, and we will email you a link to reset your password.
Forgot your password?