Bound on norm of unit vector (PSET 8, ex 2 part 1)

Hello, I have a question regarding the following bound on the norm:

if \(u = \sum_i c_i u_i\), then \(||u||^2 = \langle u, u \rangle = \langle \sum_i c_i u_i, \sum_j c_j u_j \rangle = \sum_{i, j} c_i c_j \langle u_i, u_j \rangle = \sum_i c_i^2 \) if we assume that the \( \{u_i\} \) forms an orthonormal basis. Hence I don't understand why there is an inequality in the solution. Where is the mistake in my reasoning ? (Note: \(c_i\) is the projection of \(u\) on \(u_i\) in the solution)

Thank you for your help
Justin

Yes, it should be equal. I forgot to correct it.

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