has a solution Show that f must be orthogonal to v for every

has a solution. Show that f must be orthogonal to v for every v satisfying

Solution

SOLITION:

Let T be a linear operator in a inner product space V.The adjoint of T is denoted byT* is a linear operator on V wich satisfies,

<x,Ty>=<T*x,y> for all x,y in V.

Here l is linear operator and lu=f

Now, <v,f> = <v,lu> = <l*v,u> = <0,u> = 0

Therefore f is orthogonal to v for every v satisfying l*v =0