In the Dirichlet eigenvalue problem $\Delta u-\lambda u=f$ in $\Omega$ and $u=0$ on $\partial\Omega$. I need to show if $\lambda=\lambda_k$ for some $k$ then tbe above equation has unique solution iff $\int f\phi=0$ for all $\phi\in B_k$ where $B_k$ is space of eigenfunction associated to the eigenvalue $\lambda_k$.
One side I have proved. I could not able to prove that if $\int f\phi=0$ then there exists a unique solution when $\lambda=\lambda_k$;