Please give examples of countably-infinite matrices that are Hermitian &/or invertible, & that correspond to operators commonly encountered in physics

Question

I'm new to quantum mechanics, and I believe that understanding infinite vector spaces should help my understanding of the math involved. Most of the time I'm only dealing with infinitely differentiable functions that also have a Taylor Series expansion, and I believe this implies that {$1, x^2, x^3, x^4, ... $} serves as a basis of cardinality $|\Bbb{N}|$ for the vector space of all such functions, and that therefore even though these functions live in a $|\Bbb{R}|$-dimensional space, they can still be restricted to a $|\Bbb{N}|$-dimensional space, similar to how vectors restricted to a 2D plane also can live in a 3D space. Please give examples of countably-infinite matrices that are Hermitian &/or invertible, & that correspond to operators commonly encountered in physics. Thanks!