Orthogonal complement in $ L_2 $ Find the orthogonal complement of a subspace
$$ M = \{ x \in L_2(-1, 1):x(t)=-x(-t), \int_0^1 x(t)t^2dt=0 \} $$
in $L_2(-1, 1).$
As I understand M can be described as all odd functions which are orthogonal to $ \lambda t^2 $ on $(0, 1)$. But I don`t know how to find the orthogonal complement

Let $f(t)=t^{2}\chi_{(0,1)}(t)$. Then $M$ consists of functions which are odd and orthognal to $f$. This means $M$ is precisely the orthogonal complement of the span of even functions and $f$. Thus, $M^{\perp} =\{g+cf:c \in \mathbb R, g \, \text {is even} \}$
