Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 58 / 1 / a

Let $\{|u_j⟩{\}}_{j=1}^N$ be another complete orthonormal basis, and write $|u_j⟩={\sum }_n{U}_{nj}|n⟩$. Orthonormality makes $U$ a unitary matrix, so $U{U}^{†}=I$. ThereforeThus the resolution of the identity is independent of the orthonormal basis. Equivalently, expanding any vector in the new complete basis shows that the sum of its orthogonal projections returns that vector.