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holezch
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Homework Statement
V: complex inner product space with adjoint T*
Suppose that < T( x ) , x > = 0 for all x in V, then T is the zero transformation.
The Attempt at a Solution
< T( x ) , x > = < x, T*(x ) > = 0
0 = < x, 0 > = < 0, x >
< x, T*(x ) > = 0 = < x, 0 >
if < x , y > = < x, z> , then y = z
so T*(x ) = 0 for any x, which means T* is the zero transformation, which implies that T is the zero transformation..
is this okay? thanks