In Schwartz's distribution theory, $T_\nu\rightarrow 0$ in $\mathscr{D}'(\mathbb{R}^n)$ is defined to be $\langle{T_\nu,\varphi\rangle} \rightarrow 0,$ $\forall \varphi\in C_c^\infty(\mathbb{R}^n),$ and $T_\nu(B)\rightarrow 0$ for all bounded set $B$ in $\mathscr{D}.$
Here $B$ is a bounded set in $\mathscr{D}$ means $B$ has the form like: $$ \{\varphi\in C_c^\infty(K)|\sup_x|\partial^\alpha\varphi(x)|\le M_\alpha\} $$ for certain compact set $K,$ and $T(B):=\sup_{\varphi\in B}|\langle T,\varphi\rangle|.$
However, I can't come up with an example that $\langle T_\nu,\varphi\rangle\rightarrow 0,$ $\forall\varphi\in C_c^\infty(\mathbb{R}^n),$ but without $T_\nu(B)\rightarrow 0$ for certain bounded set $B$ in $C_c^\infty(\mathbb{R}^n).$ Is there any obvious example? Otherwise we shall delete the supplement after the word "and".
I think about these when solving my question: $T\mapsto T*\varphi$ is a continuous map . I think $T_\nu\rightarrow 0$ shall add condition that $T_\nu(B)\rightarrow 0.$ After that I can solve it.