If we are talking about functions, they are orthonormal if the integral of their products is zero if they are different functions and one if they are the same.

If we are talking about vectors, they are orthonormal if the dot product is zero if they are different functions and one if they are the same.

In each case, all of the things are orthogonal to one another, and normalized.

A subset A := (a_1, a_2, ..., a_n) of a Euclidean vector space V with inner product I is said to be orthonormal if every pair of vectors in A are orthogonal and I(a_i,a_i)= ||a_i|| = 1 for i=1,2,...,n.

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