Inner product space - minimization.

[binbagsss]

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)=^{1}_{-1} \intf(x)g(x)dx

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 \in C[-1,1].

However I am 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.

 

[haruspex]

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)?