The implications ( 1 ) ⇒ ( 2 ) {\displaystyle {}(1)\Rightarrow (2)} , ( 2 ) ⇒ ( 3 ) {\displaystyle {}(2)\Rightarrow (3)} and ( 3 ) ⇒ ( 4 ) {\displaystyle {}(3)\Rightarrow (4)} are restrictions. ( 4 ) ⇒ ( 3 ) {\displaystyle {}(4)\Rightarrow (3)} . For the zero vecto r,the statement ( 3 ) {\displaystyle {}(3)} is clear; so suppose that v ≠ 0 {\displaystyle {}v\neq 0} . Then, v ‖ v ‖ {\displaystyle {}{\frac {v}{\Vert {v}\Vert }}} has norm 1 {\displaystyle {}1} , and, because of
we have
( 3 ) ⇒ ( 1 ) {\displaystyle {}(3)\Rightarrow (1)} follows from fact.