Not sure, the functions f(x) =x^n , n positive, all start from zero and end in 1.? Or 1-x^n start from 1, end in zero.?
Your looking to convert to some orthogonal system of special functions.?
You should revise the literature to see if one meets the condition, or can be adapted. Legendre, Tchebyshev , etc Within such systems one talks about a complete set of functions and a vector space with functions with inner product.
Yes, the set of continuous functions on the closed interval [0, 1] to [0, 1], equipped with the supremum norm forms a complete metric space.
But If we equip the space of continuous functions on [0, 1] to [0, 1] with a norm that is not the supremum norm, then it may or may not be a complete metric space.