Let $$\langle f,g \rangle=f(0)\overline{g(0)}+\int_{0}^1 f'(x)\overline{g'(x)}dx$$ over $C^{1}[0,1]$ find an orthonormal system $\{h_1,h_2,h_3\}$ s.t $$Span\{h_1,h_2,h_3\}=Span\{1,x,x^2\}$$

I have started with checking $\{1,x,x^2\}$ which is not orthogonal, So the only way is to use Gram–Schmidt process?

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    $\begingroup$ Gram-Schmidt is the way to go. $\endgroup$ – humanStampedist Sep 29 '17 at 12:48
  • $\begingroup$ Gram-Schmidt is THE algorithm to this kind of computation. $\endgroup$ – Thadeu Henrique Costa Sep 29 '17 at 12:54
  • $\begingroup$ @humanStampedist The only way? $\endgroup$ – gbox Sep 29 '17 at 14:15
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    $\begingroup$ @gbox I think unless you have some prior information or intuition about what the form of the orthonormal basis should be (which might be possible, say, in the case of the Chebyshev polynomials), G-S is the only recourse. $\endgroup$ – RideTheWavelet Sep 30 '17 at 3:40

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