We define an increasing sequence of closed subspaces \begin{align*} V_{0} \subset V_{1} \subset V_{\ell} \subset \dots \end{align*} of $L^2(I)$ where $I=(0,x_{max})$, and each $V_{\ell}$ is equipped with a $L^{2}$ basis $ \{\phi^{\ell}_i\}_{i=1}^{m_{\ell}} $ ( $\phi^{\ell}_i$ are piece-wise polynomials of order $p$). I define two types of orthogonal projections
\begin{align*} P_L : & L^2(I) \rightarrow V_L\\ &f \rightarrow P_L f \end{align*} and a nested projection for $\ell\leq L$
\begin{align*} P_{\ell,\ell-1} : & V_{\ell}\rightarrow V_{\ell-1}\\ & f^{\ell} \rightarrow P_{\ell,\ell-1} f^{\ell}=\tilde{f}^{\ell-1} \end{align*}
I have a problem where I am interested on deriving an upper bound estimate for
\begin{align} \mathrel{\Big|} \mathrel{\Big|} \tilde{f}^{\ell}- \tilde{f}^{\ell-1} \mathrel{\Big|} \mathrel{\Big|}_{L^2(I)}^2. \end{align}
A tentative way I did, is the following \begin{align} \mathrel{\Big|} \mathrel{\Big|} \tilde{f}^{\ell}- \tilde{f}^{\ell-1} \mathrel{\Big|} \mathrel{\Big|}_{L^2(I)}^2 &= \mathrel{\Big|} \mathrel{\Big|}\tilde{f}^{\ell}- P_{\ell,\ell-1} \tilde{f}^{\ell} \mathrel{\Big|} \mathrel{\Big|}_{L^{2}(I)}^2 \nonumber \\ &\leq \mathrel{\Big|} \mathrel{\Big|}\tilde{f}^{\ell}- f \mathrel{\Big|} \mathrel{\Big|}_{L^{2}(I)}^2+ \mathrel{\Big|} \mathrel{\Big|} f - P_{\ell,\ell-1} \tilde{f}^{\ell} \mathrel{\Big|} \mathrel{\Big|}_{L^{2}(I)}^2 \nonumber\\ & \le \mathrel{\Big|} \mathrel{\Big|} \left( P_{\ell+1,\ell} \dots P_{L,L-1} P_{L}\right) f - f \mathrel{\Big|} \mathrel{\Big|}_{L^{2}(I)}^2+ \mathrel{\Big|} \mathrel{\Big|} f- \left( P_{\ell,\ell-1} P_{\ell+1,\ell} \dots P_{L,L-1} P_{L} \right) f \mathrel{\Big|} \mathrel{\Big|}_{L^{2}(I)}^2 \nonumber\\ & \le C h_{\ell-1}^{2p+2} \mathrel{\Big|} \mathrel{\Big|} f^{(p+1)} \mathrel{\Big|} \mathrel{\Big|}_{L^{2}(I)}^2, \end{align} where $h_{\ell-1}$ is the discretization mesh size of $V_{\ell-1}$, and where I used the interpolation error estimate as an upper bound.
I am not sure if this is correct, or if there are sharper bounds. Any hint or reference in this regard. Thanks.
Since your spaces are nested, $P_{\ell + 1,\ell}\,P_{\ell+2,\ell+1} \dots \, P_{L,L-1}$ is the orthogonal projection from $L_2(I)$ to $V_\ell$. Indeed, using the notation $P_\ell = P_{\ell + 1,\ell}\,P_{\ell+2,\ell+1} \dots \, P_{L,L-1}$, then for $f \in L^2(I)$ and for any $e \in V_\ell$, $$\langle P_\ell f - f, e \rangle = \langle P_{\ell+1,\ell}(P_{\ell+1}f) - (P_{\ell + 1}f), e\rangle + \langle P_{\ell+1}f - f,e\rangle = 0 \, + \,\langle P_{\ell+1}f - f,e\rangle,$$ and by recursion the second term is 0 too. Using this, $$\|\bar f^\ell - \bar f^{\ell-1}\|^2 \leq 2\|P_{\ell-1} f -f\|,$$ from which you can proceed by using the interpolation error as you did.