The polynomials can be given an inner product $$\int_X p(x)q^*(x)dx$$
Where X is an interval on the real number line.
Consider instead, that X is a curve over the complex field with parametrization $f(t): [0,T] \to C$
$$\int_X p(x)q^*(x)dx = \int_0^T p(f(t))q^*(f(t)) f'(t)dt$$
What kind of curves define a valid inner product over the polynomials?
Only the horizontal segments, parametrized from left to right, work for this purpose. Indeed, the quantity $\int_0^T |p(f(t))|^2 f'(t)\,dt$ must be real valued and positive, for every nonzero polynomial $p$. Polynomials, being dense among continuous functions on an interval, can approximate point mass at any point of $[0,T]$: that is, have a sharp spike near one point and be nearly zero elsewhere. Hence, $f'\ge 0$ must hold pointwise on $[0,T]$.