I have a space of interpolants given by: \begin{equation} S:= \left\{ \sum_{i \in F} c_i \eta\Bigl(\frac{t-ih}{h\sqrt{D}}\Bigr):(c_i)_{i \in F} \subset \mathbb{R} \right\}. \end{equation} where $F$ is a finite set, $h>0, D>0$. I know $S \subset L^2([0,1];\mathbb{R})$. Let $T \subset S$, which is also a finite dimensional subspace of $L^2$. Let $f,g \in T$. I know the following estimate $$\| f-g \|_2 \le \varepsilon_0$$ I want to find the estimate $\|f-g\|_{\infty}.$ To that end, I want to know that is there any way, or any known inequality, such that I can bound $\|f-g\|_{\infty}$ by $\|f-g\|_{2}$? I can not directly use the fact that norms are comparable in finite dimensions because $f,g$ are functions.
One particular way I was thinking of is using Reproducing kernel Hilbert space methods. As $T$ is a finite dim subspace of $L^2$ it is an RKHS; thus $$|f(x)-g(x)|\le \sqrt{K}\|f-g\|_{H}$$ where K is the kernel function and $\|\cdot\|_H$ is the Hilbert space norm. Is this a valid way forward? Any suggestions?