Prove that
i) π ππ₯ , ππ¦ = π d(x,y)
ii)π π + π₯ , π + π¦ = π π₯ , π¦
where d is a metric induced by on a normed space X
d(x,y)=ββ£xiβyiβ£d(x,y)=\sum |x_i-y_i|d(x,y)=ββ£xiββyiββ£
i)
d(ax,ay)=ββ£axiβayiβ£=aββ£xiβyiβ£=ad(x,y)d(ax,ay)=\sum |ax_i-ay_i|=a\sum |x_i-y_i|=ad(x,y)d(ax,ay)=ββ£axiββayiββ£=aββ£xiββyiββ£=ad(x,y)
ii)
d(a+x,a+y)=ββ£a+xiβyiβaβ£=ββ£xiβyiβ£=d(x,y)d(a+x,a+y)=\sum |a+x_i-y_i-a|=\sum |x_i-y_i|=d(x,y)d(a+x,a+y)=ββ£a+xiββyiββaβ£=ββ£xiββyiββ£=d(x,y)
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