 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2010/b/5/ii/solution

Self-adjointness and conjugate symmetry of the inner product give

⟨ T x, x ⟩ = ⟨ x, T x ⟩ = \overline{⟨ T x, x ⟩} .

A complex number equal to its conjugate is real, so $⟨ T x, x ⟩ \in R$ for every

x

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