Let an inner product be defined on $\mathcal{C}[-1,1]$ in the following manner: $$\langle f,g \rangle=\int_{-1}^{1} g(x)f(x) dx$$ It is easy to check that this is an inner product, which is $x \perp x^2$. My problem is that is there a way of finding a function $h$ that is $(h \perp x) \wedge (h \perp x^2)$? Normalizing $h$ is easy since I can divide by its norm as a scalar which carries over under the $\int$. sign.
$\textbf{Question: can I find it without using Gram Schmidt}$? I already have a solution that is $$h(x)=\frac{-15}{8}(x^2-3/5)$$ but since we haven't discussed that process in class there must be a simpler way
$$\int_{-1}^{1} xh(x) dx=\int_{-1}^{1} x^2h(x) dx=0$$
Orthogonal functions for this inner product are Legendre polynomials: [1], [2]. You'll find that besides the Gram-Schmidt procedure there are other generating formulas: Bonnet’s recursion formula, Rodrigues formula, summation formulas.