# If a basis set is complete, are the elements in it mutually orthonormal?

If a basis set is complete, are the elements in it mutually orthonormal? For example, we can express the field operator in the basis of the creation and annihilation operators.This basis is complete, are the elements in it mutually orthonormal?

Assuming the original basis is complete and orthogonal, and contains $$V_1$$ and $$V_2$$, (which are thus orthogonal).
Replacing $$V_2$$ by $$V_2'=V_1+V_2$$ does not change the completude of the basis. But now two vectors of the new basis, namelt $$V_1$$ and $$V'_2$$ are not orthogonal.