Let f : R E be injective and continuous, where (E, kk) is a Banach space. Define d: R R R by d(x, y) = kf(x) f(y)k for all x, y R. 1. (3 points) Show that (R, d) is a metric space. 2. (5 points) Show that (R, d) is complete if and only if f(R) is closed. 3. (2 points) Using the above questions, decide whether (R, ) is a complete or an incomplete metric space, where : R R R is defined by (x, y) = |e x e y | for all x, y R.

Exercise 5 ( 10 points ) . Let f : R - E be injective and continuous , where ( E , \\\\ . (1 ) is a Banach space . Define d: 1R X KR -> IR by d ( ac , y ) = 1 1.f ( 20 ) - f ( y)|1 for all x , y E R . 1 . ( 3 points ) Show that ( RR , d ) is a metric space . 2 . ( 5 points) Show that ( R , d ) is complete if and only if f ( IR ) is closed . 3. ( 2 points ) Using the above questions , decide whether ( R , p ) is a complete or an incomplete metric space , where p : 1R X KR - IR is defined by P ( ac , y ) = |ex - ey| for all x , y E R