For the following Euclidean vector spaces $(V,\omega)$ and a basis $f_1,\ldots,f_n$, run Gram-Schmidt orthonormalisation process to arrive at an orthonormal basis.
(i) $V=\mathbb R^3$ with the standard dot-product $\omega(\bf v, \bf w) = \bf v^T\bf w$; $f_1=(1,0,0)^T$, $f_2=(1,1,1)^T$, and $f_3=(1,-1,1)^T$.
(ii) $V=\mathbb R[X]_{\leqslant4}$ is the space of real polynomials of degree at most 4, $$\omega(f(X),g(X)) = \int_{-1}^1 f(X)g(X)\ \mathsf dX\; ; $$
$f_i = X^{i-1}, i=1,2,3,4,5$.
(iii) $V=\mathbb R[X]_{\leqslant4}$ is the space of real polynomials of > degree at most 4, $$\omega(f(X),g(X)) = \int_{-\infty}^{+\infty} e^{-X^2}f(X)g(X)\ \mathsf dX\; ;$$
$f_i = X^{i-1}, i=1,2,3,4,5$. (Hint: } You will need the Gaussian integral
$$\int_{-\infty}^{+\infty} X^{2k}e^{-X^2}\ \mathsf dX = 2^{-k}(2k-1)!!\sqrt\pi $$
where $(2k-1)!!$ is the double factorial defined by $n!!=n\cdot(n-2)!!$ for $n>0$ odd and $(-1)!!=1$.)
Part iii) I'm not sure if I am doing this correctly but for 2 marks this is taking me ages! So far I have $1, x-\frac{1}{\sqrt2}, x^2-\frac{1}{2}, x^3-\frac{3}{\sqrt2}x^2+\frac{1}{2\sqrt2}$ but the next one doesn't look nice. Are the ones I have computed so far correct (Yes, I know they are not orthonormal yet)