Adjoint of Inner Product Space Question

[PuzzledMath]

Homework Statement

Suppose T is in the set of linear transformations on an inner product space V and lamba is in F, the scalar field of V. Prove that lamda is an eigenvalue of T iff lamda_bar (the conjugate) is an eigenvalue of T*, the adjoint of T.

Homework Equations

Using [] brackets to denote an inner product:
The definition of the adjoint is
[Tv, w] = [v, T*w ] for all v, w in V
Some Properties of the inner product:
[v, v] >/= 0 for all v in V
[v, v] = 0 iff v = O
[v + u, w] = [v, w] + [u, w]
[av, w] = a [v, w]
[v, aw] = a_bar [v, w]
Some Properties of the Adjoint:
(T*)* = T
(aT)* = a_bar T*

The Attempt at a Solution

I've been working on the first direction of the iff statement.

We know Tx = lamda x for some x in V
Then
[Tx, w]
= [ x, T*w]
= [ lamda x, w]
= [ x, lamda_bar w] using conjugate linearity property
This is for any w in V.

Then [ x, T*w - lamda_bar w ] = 0 for any w in V.

But how can I get the conclusion that T*v = lamda_bar v for some v in V?
I've tried letting w = x, but it doesn't seem to help.
I've also tried starting with something like
[Tw, lamda x]
= [w, lamda_bar T*x]
= [Tw, Tx ]
= [w, T*(Tx)]
but I don't know what T*(Tx) is, so that doesn't help me.

My professor isn't answering email over spring break, and this is frustrating me. I keep trying and don't get anywhere. Any advice would be greatly appreciated! Thanks.

 

[micromass]

Allright, have you seen the following theorem:

If X and Y are pre-Hilbert spaces and if T:X\rightarrow Y is a bounded linear operator. If <T(x),y>=0 for all x in X and y in Y, then T=0.

Try to prove this first. Then you can apply this theorem to the space

X=Y=span\{x\}

where x is your eigenvector.

 

[PuzzledMath]

I don't have any theorem like that - I don't even know what a pre-Hilbert space is. My professor said it was a very simple, short proof with applications of definition and properties we have, but I can't figure it out. I've listed most of the information I have to work with, but I can't make it do what it's supposed to : (