The Legendre polynomials can be used to construct an infinite set $X \subset C[0,1]$ with the property that $$\int_{0}^{1} f(x)g(x) \ dx = 0$$ for any distinct $f,g \in X$.
I want to see if this can be generalized.
$\textbf{Definition:}$ For any positive integer $k$ with $k \geq 2$, let's say a linearly independent $X \subset C[0,1]$ is $k$-orthogonal if for any finite $S \subset X$ with $|S| = k$ we have that $$\int_{0}^{1} \prod_{f \in S} \ f(x) \ dx = 0$$
For any odd $k$, the (infinite) set $X=\{\sin^{2m+1}(x)\}_{m \geq 0}$ is $k$-orthogonal.
$\textbf{Problem 1}:$ Must there always exist an infinite $k$-orthogonal set for even $k$?
It's possible that some slight modification of the odd case can yield a solution, but I have not been able to make it work.
$\textbf{Definition:}$ Let's say an infinite, linearly independent set $X \subset C[0,1]$ is completely orthogonal if for every finite $S \subset X$ with $|S| \geq 2$ we have that $$\int_{0}^{1} \prod_{f \in S} f(x) \ dx = 0$$
$\textbf{Problem 2:}$ Does there exist a completely orthogonal set $X \subset C[0,1]$?