Inner product space - minimization.

  1. 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.
    Last edited: Apr 6, 2013
  2. jcsd
  3. haruspex

    haruspex 16,047
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    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)?
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