quantum mechanics - Eigenvalues of Hermitian operators are real and the dependence/independence of boundary conditions

Without reproducing proofs:

1. 
   

   Eigenvalues of a Hermitian operator are real (proof does not rely on the boundary conditions).

   
   
   


2. 
   

   The momentum operator is Hermitian (proof does not rely on the boundary conditions).

   
   


3. 
   

   Without any boundary conditions, eigenvalues of the momentum operator can be complex.

   
   

How is this possible? Proofs of 1, 2, and 3 can be found in most introductory texts, but I can reproduce them if their veracity is in question.

Added: By Hermitian I mean $\int f^* (D g) \, d^3r=\int (Df)^* g\, d^3r$. The proof of 2 rely on $fg\to 0$ as $x\to\infty$. The boundary conditions that force eigenvalues of the momentum operator to be real is the periodicity of the eigenfunction.