What is $L^2(\Omega)/\mathbb R$?

Question

I've encountered this space in a book (see Proposition 1.2) and don't understand what is meant. It's a notation I only know from quotient spaces, but I can't make sense of that here.

Answer 1

On page 11 in your link it is written that this is the space of all function orthogonal to the constant functions,

$$\{p\in L^2(\Omega) :\int_\Omega p\, dx = 0\}$$

Answer 2

Consider the Stokes (and similarly Navier-Stokes) problem in $\Omega$ \begin{equation} \begin{cases} -\mu\Delta u + \nabla p & = f,\\ \nabla \cdot u & = 0, \end{cases} \end{equation} with a presecribed boundary condition on $u$. For any solution $(u,p)$ of the problem and any $\bar{p} \in \mathbb{R}$, $(u, p+\bar{p})$ is also a solution. Therefore the solution (if exists) is unique up to an isomorphism in $L^2(\Omega)$. The solution space $L^2(\Omega)$ for pressure is thus decomposed into a direct sum $$L^2(\Omega) = \mathbb{R} \oplus (L^2(\Omega)/\mathbb{R}),$$ and the average-removed pressure is then uniquely determined in the space $$L^2(\Omega)/\mathbb{R} = \{q \in L^2(\Omega): (q,1)_{0,\Omega} = 0\}.$$