# Inner product space - minimization.

1. Apr 5, 2013

### 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 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.

Last edited: Apr 6, 2013
2. Apr 5, 2013

### 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)?