I was wondering how to eigendecompose the derivative operator.
We know that, using the monomial basis, the derivative operator has countably finitely many dimensions.
At the same time, it has uncountably infinitely many eigenfunctions, of the form $ce^{\lambda x}$ where $c, \lambda$ are arbitrary real numbers. All of these eigenfunctions are analytic, so they should be representable using the monomial basis.
This implies that the matrices the derivative operator is decomposed into has uncountably infinitely many dimensions. Which makes no sense.
How would I do eigendecompose the derivative operator in the monomial basis? Is that even possible?
When you say "using the monomial basis" you are missing something. The monomials are a basis of what?
They are not an algebraic basis of the space of continuous functions. They span a dense subspace of that space, but not every continuous function is a polynomial. The true dimension of the space of continuous functions is uncountable infinity.