Question

Every mertric space is not normed linear space give example

Answer 1

Let VV be a vector space over the field FF. A norm

∥⋅∥:V⟶F‖⋅‖:V⟶F

on VV satisfies the homogeneity condition

∥ax∥=|a|⋅∥x∥‖ax‖=|a|⋅‖x‖

for all a∈Fa∈F and x∈Vx∈V. So the metric

d:V×V⟶F,d:V×V⟶F,

d(x,y)=∥x−y∥d(x,y)=‖x−y‖

defined by the norm is such that

d(ax,ay)=∥ax−ay∥=|a|⋅∥x−y∥=|a|d(x,y)d(ax,ay)=‖ax−ay‖=|a|⋅‖x−y‖=|a|d(x,y)

for all a∈Fa∈F and x,y∈Vx,y∈V. This property is not satisfied by general metrics. For example, let δδ be the discrete metric

δ(x,y)={1,0,x≠y,x=y.δ(x,y)={1,x≠y,0,x=y.

Then δδ clearly does not satisfy the homogeneity property of the a metric induced by a norm.

To answer your edit, call a metric

d:V×V⟶Fd:V×V⟶F

homogeneous if

d(ax,ay)=|a|d(x,y)d(ax,ay)=|a|d(x,y)

for all a∈Fa∈F and x,y∈Vx,y∈V, and translation invariant if

d(x+z,y+z)=d(x,y)d(x+z,y+z)=d(x,y)

for all x,y,z∈Vx,y,z∈V. Then a homogeneous, translation invariant metric dd can be used to define a norm ∥⋅∥‖⋅‖ by

∥x∥=d(x,0)‖x‖=d(x,0)

for all x∈Vx∈V.