The question is : If the vector space C[1,1] of continuous real valued functions on the interval [1,1] is equipped with the inner product defined by (f,g)=[itex]^{1}_{-1}[/itex] [itex]\intf(x)g(x)dx[/itex] Find the linear polynomial g(t) nearest to f(t) = e^t? So I understand the solution will be given by (u1,e^t).||u1|| + (u2,e^t).||u2|| But I am having trouble understanding what u1 and u2 should be. I understand they must be othorgonal and basis for a subspace S [itex]\in[/itex] C[-1,1]. However Im not too sure what dimension this basis should be of, and not 100% sure what is meant by the vector space C[-1,1]. (The solution uses 1 and t as u1 and u2....) Many thanks in advance for any assistance.
I think you're making it overcomplicated. Isn't it just asking for the affine function g(t) which minimises the integral for the given f(t)?