I’m always on the lookout for interesting problems, particularly ones that I might assign my students at some point. The content of this post comes from Exercise 19, Chapter 4 of Rudin’s *Real and Complex Analysis*.

Fix a positive integer , and set . Then we have the orthogonality relations

*Proof.* It is clear that , for . For the second assertion, observe that and by the geometric series formula,

In the case , we can use the orthogonality relations to derive the following identity for inner product spaces:

Similarly, for any inner product space, we have the identity

We can also obtain the preceding identity from the preceding identity and the dominated convergence theorem. Indeed,

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