Earlier this day, I asked a question , on how to prove continuity of scalar multiplication and addition in a specific topological vector space. To repeat it here, I was talking about this space:
Let $\mathcal{D}_K$ be the space of all on $K\subseteq \mathbb{R}$ compactly supported, infinitely differentiable functions.
I have shown for $N \in \mathbb{N}$, $\epsilon > 0$ and $U_{N,\epsilon} := \{f \in \mathcal{D}_K: \max_{0\leq i \leq N} > ||f^{(i)}||_\infty < \epsilon\}$, that the sets $f + U_{N,\epsilon}$ with $f\in \mathcal{D}_K$ form a basis of a topology on our space.
I do now indeed understand how to prove that this space is a topological vector space. However, I also want to show that the differentiation operator $D_i: \mathcal{D}_K \rightarrow \mathcal{D}_K , f \mapsto f^{(i)}$ is continuous under this topology. As far as I have understood, this topology is induced by the norms $$||g||_N := \max_{i \leq N} ||g^{(i)}|| $$ so essentially my open sets are just unions of balls $B_N(g,\epsilon)$, with some norm $||\cdot||_N$
I now tried to use this to prove the continuity of $D_i$, by showing that its preimage of such an open ball $$D_i^{-1}(B_N(f,\epsilon)) = \{g \in \mathcal{D}_K : \max_{k\leq N} ||g^{(i+k)}-f^{(k)}||_\infty < \epsilon \} $$ is contained in a union of other open balls containing $f$. This is where I get stuck. I always end up with derivatives of different order on each of my functions, so I do not know how I can find such a union. Any help would be greatly appreciated!
Let $f_\alpha \to f$ in $\mathcal D_K$. This is equivalent to $\|f^{(k)}_\alpha - f^{(k)}\|\to0$ for all $k$. It follows that $$\|(D_if_\alpha)^{(k)}-(D_i f)^{(k)}\|=\|f^{(i+k)}_\alpha-f^{(i+k)}\|\to0$$ for all $k$ and thus $D_i f_\alpha\to D_i f$ in $\mathcal D_K$. This verfies the continuity of the map $D_i$.