Hermitian Operators and Non-Orthogonal Bases: Exploring Infinite Spaces

[LCSphysicist]

TL;DR

It is not necessary.

The basis he is talking about: {1,x,x²,x³,...}
I don't know how to answer this question, the only difference i can see between this hermitians and the others we normally see, it is that X is acting on an infinite space, and, since one of the rules involving Hermitian fell into decline in the infinity space (e.g Eigenvectors span the space),i am not sure if another rules changes in the infinity space too..

 

[Infrared]

The matrix of a self-adjoint operator with respect to an orthonormal basis is equal to its conjugate transpose. Your basis is not orthonormal.

 

[LCSphysicist]

The matrix of a self-adjoint operator with respect to an orthonormal basis is equal to its conjugate transpose. Your basis is not orthonormal.

Forgot this detail, thank you.