Consider a family of Banach spaces $(\mathcal{B}_{t},\Vert\cdot\Vert_{t})_{t\in\mathbb{R}}$ and define the space $C^{\infty}(\mathbb{R},\mathcal{B}_{\bullet})$ where smootheness has to be understood in the sense of the Frechét (total) derivative. Now, let $I\subset\mathbb{R}$ be a compact interval. We set
$$\Vert f\Vert_{I}:=\int_{I}\Vert f(t)\Vert_{t}\,\mathrm{d}t$$
for all $f\in C^{\infty}(\mathbb{R},\mathcal{B}_{\bullet})$.
Is the space $C^{\infty}(\mathbb{R},\mathcal{B}_{\bullet})$ a Frechét space with respect to topology induced by the family of seminorms $(\Vert \cdot\Vert_{I})_{I\subset\mathbb{R}}$?
My idea: First of all, the family of seminorms can be reduced to a countable one by taking a compact exhaustion of $\mathbb{R}$. Secondly, we can estimate $$\min_{t\in I}\Vert f(t)\Vert_{t}\leq \Vert f\Vert_{I}\leq\max_{t\in I}\Vert f(t)\Vert_{t},$$ which maybe can be used to argue that $C^{\infty}(\mathbb{R},\mathcal{B}_{\bullet})$ is complete.