Answer to Question #272131 in Functional Analysis for Prathibha Rose

Question

Prove that

i) 𝑑 𝑎𝑥 , 𝑎𝑦 = 𝑎 d(x,y)

ii)𝑑 𝑎 + 𝑥 , 𝑎 + 𝑦 = 𝑑 𝑥 , 𝑦

where d is a metric induced by on a normed space X

Answer 1

$d(x,y)=\sum |x_i-y_i|$

i)

$d(ax,ay)=\sum |ax_i-ay_i|=a\sum |x_i-y_i|=ad(x,y)$

ii)

$d(a+x,a+y)=\sum |a+x_i-y_i-a|=\sum |x_i-y_i|=d(x,y)$