What is the importance of complete orthonormal set of functions in quantum mechanics?

In quantum mechanics, a complete orthonormal set of functions is crucial because it allows for the representation of quantum states as linear combinations of basis functions, facilitating calculations and analysis. This completeness ensures that any state in the Hilbert space can be expressed as a sum of basis states, while orthonormality guarantees that the inner product of different states yields zero, simplifying the evaluation of probabilities and expectation values. Furthermore, it underpins the mathematical framework of quantum mechanics, ensuring that physical observables can be accurately measured and predicted.