Consider the inner product space
$$C_{[-1,1]}\text{ with inner product :=}\int ^1_{-1} f(x)g(x)dx$$ Let $U=\{\frac{1}{\sqrt{2}},\frac{\sqrt{6}}{2}x\}$ form basis for a subspace of C, than
1.Show that $U$ is an orthogonal basis . and
2.Find the best least square approximation to $h(x)=x^\frac{1}{3}+x^\frac{2}{3}$ by linear function.
My attempt as $\|f\|=\|g\|=1 $ and $<f,g>=0$ than we can say that $U$ is an orthonormal basis
How to solve the second question any hint, please
How to solve
Since $\|f\|=\|g\|=1$ and since $\langle f,g\rangle=0$, $\{f,g\}$ is an orthonormal basis of the space of all first degree polynomials. So, take $\langle h,f\rangle f+\langle h,g\rangle g$.