Let $\mathcal{E}(\mathbb{R})$ be the space of all $C^\infty$ functions on $\mathbb{R}$ with its usual topology, and $\mathcal{E}'(\mathbb{R})$ be the dual space with the weak* topology.
Let $(T_i)_{i\in I}$ be a net in $\mathcal{E}'(\mathbb{R})$ that converges to $T$ in $\mathcal{E}'(\mathbb{R})$.
Is $(T_i)_{i\in I}$ bounded (in the Topological Vector Space sense of absorption by dilations of neighbourhoods of $0$)?
Not in general. For example let $I_0$ be an arbitrary set and $(T_i)_{i\in I_0}\ \ $ be an arbitrary family of elements of $\mathcal E'$. Let $I=I_0\cup\{\alpha\}\ \ $ for some new element $\alpha$ and define a partial order on $I$ by setting $\alpha>i$ for any $i\in I_0$. Then $I$ is a directed set. Define a net by extending the given family by $T_\alpha=0$. This net converges to zero, actually, it is eventually stationary. But as the family $(T_i)_{i\in I_0}\ \ $ is arbitrary, it need not be bounded.
