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Year 3 of the mathematics course of the University of Oxford

Analysis course of the University of Oxford

Past exam of the analysis course of the University of Oxford

2011

By the definition of the <adjoint operator> and conjugate symmetry,
$$
\langle T^*x,x\rangle=\langle x,Tx\rangle
=\overline{\langle Tx,x\rangle}.
$$
Taking real parts proves
$$
\boxed{\operatorname{Re}\langle T^*x,x\rangle
=\operatorname{Re}\langle Tx,x\rangle\geq0.}
$$

Solution

Past exam of the analysis course of the University of Oxford / 2011 / 6 / b / iii / Solution

Past exam of the analysis course of the University of Oxford / 2011 / 6 / b / iii / Solution

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Past exam of the analysis course of the University of Oxford / 2011 / 6 / b / iii / Solution

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