Is there a representation of an inner product where monomials are orthogonal?

There are plenty of examples of inner products on special sequences of polynomials such that they are orthogonal. I can't quite wrap my head around the inner product s.t. monomials are orthogonal. Say we have polynomials defined on the unit interval $[0, 1]$. I can define an inner product by stating: $$\langle x^m, x^n\rangle = \delta_{mn}$$ This then extends via linearity to a full inner product on the set of all polynomials on $[0,1]$.

I can't see how this inner product can be represented with respect to Lesbesgue measure however. If there was an $h$ s.t. $$\langle f, g\rangle = \int_0^1 f(x)g(x)h(x)dx$$ then $$\langle x^m, x^n\rangle = \int_0^1 x^mx^nh(x)dx = \langle x^{m+n}, 1\rangle$$ which can't satisfy the orthogonality requirements.

My question then is, does there exist a measure (maybe a discrete one) where this inner product has a representation wrt? (or even just a formula of some kind to make it less abstract).

Solution 1:

If such a measure existed, then by your argument it follows that $$0 = \langle x^3, x^1 \rangle = \langle x^4,1 \rangle = \langle x^2 , x^2 \rangle = 1$$ which is a contradiction. Therefore such a measure can not exist.

Solution 2:

Such a measure does not exist. Polynomials that are orthogonal with regards to a positive measure (you need a positive measure to get an inner product), must have simple roots inside the support of the measure (see for example this thread).

However, there's a simple formula for your inner product : if $P = \sum_{n \ge 0} a_n X^n$ and $Q = \sum_{n \ge 0} b_n X^x$, then

$$\left\langle P,Q \right\rangle = \sum_{n \ge 0} a_n b_n$$