Suppose alpha epsilon L V V where V is an inner product spac

Suppose alpha epsilon L (V, V). where V is an inner product space. Then for its adjoint alpha*. show, that alpha is subjective if and only if alpha* is injective:alpha a is injective if and only if alpha* is surjective.

Solution

solution:

a)

since

(null a)=range a*

and null(a)={0}

if and only if a*=V

THEREFORE

a is injective if and only if a* is surjective

b)

since (range a)=null(a*)

(range(a))=V if and only if null(a*)=0

there fore

a is surjective if and only if a* is injective