Is taylor series also an orthogonal projection of a infinitely differentiable function on some subspace?

Question

I'm wondering if there exists some inner product $\langle \cdot,\cdot \rangle$ defined on all real infinitely differentiable functions such that $$1, x, x^2, x^3, \ldots$$ are orthonormal w.r.t this inner product?

If there does exist such an inner product, denote $$U_j=\text{span}(1,x,\ldots,x^j).$$ Then, is it true that, for an arbitrary infinitely differentiable function $f$, $P_{U_j}(f)$ is $f$'s $j$th order taylor expansion at $x=0$?

Answer 1

It's easy to be explicit about an inner product making the monomials orthonormal:

Lemma. If $(c_n)$ is a sequence of complex numbers with $\sum|c_n|^2<\infty$ then the power series $\sum c_nx^n$ has radius of convergence at least $1$.

Proof: Suppose $|x|<1$. Then $\sum(|x|^n)^2<\infty$, hence $\sum c_nx^n$ converges, by Cauchy-Schwarz.

So we can let $H$ be the space of power series $\sum c_nx^n$ with $\sum|c_n|^2<\infty$ and define $$\left\langle\sum c_nx^n,\sum b_nx^n\right\rangle=\sum c_n\overline{b_n}.$$

And now it follows that the orthogonal projection onto the span of the first $n$ monomials is given by the $n$th partial sum.