# The range of the self-adjoint operator

Let $A\in B(H)$($B$ is the space of linear bounded operator,H is the Hilbert space) is self-adjoint and Ker(A)={0},I want to prove that $\overline{R(A)}=H$(R(A) is the range of the operator A)

-

1. $\ker(A)=R(A^*)^\perp$
2. For a subspace $W\subset H$ we have that $W^\perp =\{0\}$ if and only if $W$ is dense.