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Consider the vector space $\Bbb{V}=P_3(\Bbb{R})$ of the real polynomials of degree less or equal 3, with the inner product given by $$\langle f,g\rangle=\int_0^1f(t)g(t)dt,\forall f,g\in\Bbb{V}.$$ Find a orthonormal basis for $[5,1+t]^{\perp}.$
I've some ideas of how to do it by finding the expression for a vector from $[5,1+t]$ and then yielding a basis for $[5,1+t]^{\perp}$ and so using the Gram-Schmidt Orthonormalization Process. But I'd like to know if there is some way of avoid this, saving effort. Has anybody some hint?

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Either apply Gram-Schmidt to $\{1,x,x^2,x^3\}$ or search through some known http://mathworld.wolfram.com/OrthogonalPolynomials.html.

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