Projection in a function space

An advanced study group compares two ways of approximating h(t)=t²by a polynomial a+bt on the interval[−1,1]. The exercise defines the inner product as the integral of f(t)g(t) over that interval and asks for the orthogonal projection onto the span of1and t. One learner substitutes the point t=0and declares the best approximation is zero. Another assumes the two coefficients are both averages without constructing the inner products. The task is to derive the projection condition, use parity correctly and verify the residual. The problem concerns a finite-dimensional subspace of the specified inner-product space, not interpolation at a chosen pair of points or a statement about every possible approximation criterion.