Random variables in weak$^*$-dual of Banach space Let $X$ be a separable Banach space with weak$^*$ dual $X'$ and $T_n, T$ random variables in $X'$ (with the Borel $\sigma$-algebra). Assume that $T_n x \to T x$ in distribution for all $x \in X$. Then clearly for each $x \in X$ the sequence $T_n x$ is bounded in probability (= tight), i.e. $\forall \varepsilon > 0 \, \exists C_{\varepsilon, x} > 0: \, \sup_n P(|T_n x| > C_{\varepsilon, x}) < \varepsilon$. Does it follow that the sequence of dual norms $\lVert T_n \rVert$ is bounded in probability as well, i.e. $\forall \varepsilon > 0 \, \exists C_{\varepsilon} > 0: \, \sup_n P(\lVert T_n \rVert > C_{\varepsilon}) < \varepsilon$?
 A: No, this is not true. The following example is essentially the one in http://www.numdam.org/item/10.5802/aif.249.pdf, Exemple I.6.4.
Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space (for example $\mathcal{H} = \ell^2$).
Choose an orthonormal basis $(e_n)_{n \in \mathbb{N}}$ and define $T_n$ via
$$T_n x = \langle n  e_{J^{(n)}}, \,x \rangle,$$ where $J^{(n)}$ is a $\mathbb{N}$-valued random variable with the distribution
$$
\mathbb{P}(J^{(n)} = j) = p_j^{(n)}
$$ to be specified.
First observe that $\Vert T_n \Vert = n$. Let us now see that we can choose $J^{(n)}$ such that $\lim_{n \to \infty} \mathbb{P}(\vert T_n x \vert > \epsilon) = 0$ for all $x \in \mathcal{H}$ and all $\epsilon > 0$.
Let $$x = \sum_{j} x_j e_j \in \mathcal{H}$$ be given and $c := \sum_{j} \vert x_j\vert^2 <\infty$.
Now,
$$
\mathbb{P}(\vert T_n x \vert > \epsilon) = \mathbb{P}(\vert x_{J^{(n)}} \vert^2 > \epsilon^2 / n^2).
$$
As $\sum_j \vert x_j\vert^2 = c$, there are at maximum $\lceil c n^2 / \epsilon^2 \rceil$ of the $j$ such that $\vert x_j\vert^2 > \epsilon^2/n^2$.
If the probabilities $p_j^{(n)}$ are non-increasing, it follows that
$$
\mathbb{P}(\vert T_n x \vert > \epsilon) = \mathbb{P}(\vert x_{J^{(n)}} \vert^2 > \epsilon^2 / n^2) \leq \sum_{j = 0}^{\lceil c n^2 / \epsilon^2 \rceil - 1} p_j^{(n)} 
$$.
To ensure convergence, we now just need to choose good $p_j^{(n)}$.
You can either follow the link above or try it yourself (with for example a geometric distribution).
