Let us consider the circle $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ and for any $g \in L^2(\mathbb{T},\mathbb{C})$, define $P_N : L^2(\mathbb{T},\mathbb{C}) \to L^2(\mathbb{T},\mathbb{C})$ as \begin{equation} [P_Ng](x)=\sum_{k=-N}^N \langle g, e_k\rangle_{L^2} e_k(x) \end{equation} where $e_k(x):=e^{2\pi ikx}$. That is, $P_N$ is the Fourier projection operator.
Now, consider the Sobolev space $H^1(\mathbb{T},\mathbb{C})$ and its dual $H^{-1}(\mathbb{T},\mathbb{C})$, with the dual pairing $\langle , \rangle_{H^{-1} \times H^{1}}$
Then, I wonder how I can make sense of $P_N$ as a projection on $H^{-1}(\mathbb{T},\mathbb{C})$ so that \begin{equation} \langle P_N f, \phi \rangle_{H^{-1} \times H^{1}}=\langle f, P_N \phi \rangle_{H^{-1} \times H^{1}} \end{equation} for $f \in H^{-1}$ and $\phi \in H^{1}$.
Could anyone please explain for me?