 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2010/b/6/a/iii/solution

Apply continuity of the inner product to the convergent expansion of

x

⟨ x, y ⟩ = n \in Z \sum ⟨ x, e_{n} ⟩ ⟨ e_{n}, y ⟩ .

\sum_{n} ∣ ⟨ x, e_{n} ⟩ ⟨ e_{n}, y ⟩ ∣ \leq (\sum_{n} ∣ ⟨ x, e_{n} ⟩ ∣^{2})^{1/2} (\sum_{n} ∣ ⟨ y, e_{n} ⟩ ∣^{2})^{1/2} \leq ∥ x ∥∥ y ∥.

This is the polarized Parseval identity.

 New to topics? Read the docs here!