Let $\Omega$ be a polygonal domain in $\mathbb{R}^2$, $V$ a closed and finite dimensional subspace of $H^1(\Omega)$ and $\mathbb{P}_k(\Omega)$ the usual space of polynomials of maximum degree $k$.
I'll define the operator $\Pi : V \to \mathbb{P}_k(\Omega)$ over all the elements $v \in V$ as:
\begin{equation} \label{eq1} \tag{1} \begin{cases} a(\Pi v, p) = a(v,p) \qquad \forall p \in \mathbb{P}_k(\Omega) \\[6pt] \displaystyle \int_{\partial \Omega} \Pi v= \int_{\partial \Omega} v \end{cases} \end{equation}
where $a(\cdot,\cdot)$ is the $L^2$ inner product of the gradients: $$a(v,w) := \int_\Omega \nabla v \cdot \nabla w$$
The text I'm reading says that \ref{eq1} defines $\Pi$ only up to a constant, therefore we have to introduce another projection operator $P_0 : V \to \mathbb{P}_0(\Omega) $ to fix that constant requirring for all $v \in V$ to satisfy $$P_0 (\Pi v − v ) = 0$$
I can't get the need of $P_0$, or better, I can't understand why is there a constant to be fixed. Can anyone help me please?
EDIT
added the screenshot of the source where I found the "definition" \ref{eq1}:
