Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.

Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( I_j ) \}$,

where $\mathcal P_1$ is the space of piecewise affine functions. Let

$X_h = X_{DC} \cap C^0([0,1])$.

Given $\Theta \in X_{DC}$ with $\int_0^1\Theta = 0$, let $\theta \in X_h$ be the orthogonal projection of $\Theta$ in the energy scalar product, $(u,v) \mapsto \langle u', v' \rangle_{L^2}$. In other words,

$\forall v \in X_h : \int_0^1 \theta' v' = \int_0^1 \Theta' v'$

where $\Theta'$ is understood piecewise. Let $\|P\|$ be the linear operator that maps $\Theta$ to $\theta$.

Can you give a bound for $P$ that does not depend on $n$?