Hermitian operators without considering them as Matrices

[Master J]

A Hermitian matrix is a square matrix that is equal to it's conjugate transpose.
Now let's say I have a Hermitian operator and a function f:

[ H.f ]

The stuff in the square is the complex conjugate as the functions are in general complex. If I do not consider the matrix representation of the function and Hamiltonian, and just consider them functions, then how do I motivate the result that the above equals:

[ f ].H

I have seen this before but I am a bit confused.
Any enlightenment?

 

[HallsofIvy]

The more general concept is "self- adjoint". A linear transformation from an inner product space to itself is "self-adjoint" if and only if <Au, v>= <u, Av> where u and v are vectors in the vector space and < , > is the inner product. A real valued square matrix is "self-adjoint" if and only if it is symmetric (it is equal to its transpose) and a complex valued square matrix is "self adjoint" if and only if it is equal to its conjugate transpose.