I try to find a short method to prove that the set of all regular points $\rho(A)$ of a self-adjoint operator $A\in$ $\mathscr{B}(H)$ is an open set. Though it is true in any linear bounded operator in Hilbert space $H$ by routine check of openness, here more specific condition of " Self-Adjoint" is given.
The definitions are as follows:
A number $\lambda\in\mathbb{C}$ is called regular point iff $A-\lambda$$I$ is invertible, where $A\in$ $\mathscr{B}(X)$, $X$ is Banach space and $\overline{D(A)}$ = $X$.
Set of all regular points of $A$ is resolvent set of $A$ denoted by $\rho(A)$.