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**charikaar** Let $\displaystyle C[0,\pi]$ be the complex inner product space of complex continuous functions on $\displaystyle [0,\pi]$, with inner product

$\displaystyle \langle f , g \rangle = \int_0^{\pi} f(x) \overline{g(x)} \, dx. $

Find an orthogonal basis for the subspace of $\displaystyle C[0,\pi]$ spanned by the functions $\displaystyle {1,2ix,x^3}$

I think I have to used Gram-Schmidt method but I am not sure where to start. Can you please show me the general method and I will then be able to solve it.

Many thanks