I want to ask if the trace operator for sobolev spaces ($\gamma$) and the projection operator related to a basis commute.
So let $u\in H^2(\Omega)$ where $\Omega$ is assumed to have $C^\infty$ boundary. The boundary condition is just to gurantee sufficient regularity. The trace operator is defined as a map
$$\gamma: H^2(\Omega)\rightarrow H^1(\partial\Omega)$$
Let $\lbrace \psi_n\rbrace_{n\in\mathbb{N}}$ be a basis for $H^2(\Omega)$. Then the projection operator ($P_M$) is defined as
$$P_Mu=\sum_{j=-M}^M c_j\psi_j.$$
Now for $v\in L^2(\Omega)$ the following is given,
$$\int_{\partial\Omega}\gamma u\gamma v=\int_\Omega f v.$$
My question is this: is the following statement correct?
$$P_M(\int_{\partial\Omega}\gamma u\gamma v)=P_M(\int_\Omega f v)\tag{**}\\\int_{\partial\Omega}\gamma (P_Mu)\gamma (P_Mv)=\int_\Omega f (P_M)v),$$
If this is incorrect can some one please help in providing details what (**) should be equal to. My eventual goal is to find $\int_{\partial\Omega}\gamma (P_Mu)\gamma (P_{M^\bot} v)$.
Thanks for any help.