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A proof that for a self adjoint linear operator T, (Tf,f) is real for all f in the Hilbert space H

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Prove that for any self-adjoint bounded linear operator T on a Hilbert space H that

(Tf,f)

is real-valued for all f in H.

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Solution Summary

It is proven that for a self adjoint linear operator T, (Tf,f) is real for all f in the Hilbert space H, where (f,f) denotes the inner product on H.
The solution is detailed and well presented.

Solution Preview

Recall that if T:H to H is a bounded linear operator on the Hilbert space H then there
exists a unique and bounded linear operator T^* such that

T^* : H to H

and for all g ...

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