The couterpart of result (1) can fail if the sequence is replaced by the net. Our counterexample is based on Nate Eldredge’s counterexample. Direct a set $I=I’\times\Bbb N$ by the preorder $\preceq’$ defined by
$$(U’,n’) \preceq’ (V’, m’) \mbox{ iff } U’ \preceq V’ \mbox{ and } m’\ge n’.$$
For each $U\in\mathcal U$ pick $x_U\in X$ such that $\|x_U\|=1$ and $\langle f_U, xU\rangle\ne 0$. Define nets indexed by $I’$ putting $x^*_{(U,n,n’)}=f_{U,n}=nf_U$ and $x_{(U,n,n’)}=\frac 1{n’}x_U$ for for each $(U,n,n’)\in I$. Clearly, the net $\{ x_{(U,n,n’)}: (U,n,n’)\in I’\}$ converges to the zero. Since the net $\{f_{U,n}:(U,n)\in I\}$ converges to the zero, the net $\{ x^*_{(U,n,n’)}: (U,n,n’)\in I’\}$ converges to the zero too. On the other hand, for each $(U,n,n’)\in I’$ and each natural $m$ we have $(U,n,n’)\preceq’ (U,m,n’)$ and $\langle x^*_{(U,m,n’)}, x_{(U,m,n’)}\rangle=\langle mf_U, \frac 1{n’}x_U \rangle=
\frac {m}{n’} \langle f_U, x_U \rangle$, which has an absolute value bigger than $1$ for a sufficiently big $m$.
The couterpart of result (1) holds when the directed set $(I,\le)$ of the net has countable cofinalty, that is there exists a countable set $D$ of $I$ such that for each $n\in I$ there exists $d\in D$ with $d\ge n$. Indeed, suppose to the contrary that $\langle x^*_n, x_n\rangle\not\rightarrow \langle x^*,x\rangle$.
Then there exists $\varepsilon>0$ such that for each $n\in I$ there exists $n’\ge n$ such that
$|\langle x^*_n, x_n\rangle - \langle x^*,x\rangle|\ge\varepsilon$.
Let $\{d(k):k\in\Bbb N\}$ be any enumeration of the set $D$.
Then by indution we can build a sequence $\{n(k):k\in\Bbb N\}$ of elements of $I$ such that for each $k$ we $n(k)\ge d(k)$ and $|\langle x^*_{n(k)}, x_{n(k)}\rangle - \langle x^*,x\rangle|\ge\varepsilon$. But a sequence $\{x_{n(k)}\}$ converges to $x$ and a sequence $\{x^*_{n(k)}\}$ converges to $x^*$, a contradiction with result (1).
Best Answer
If $x_n^{*} \to x^{*}$ in weak* topology and $x_n \to x$ in the norm then $$|x_n^{*}(x_n) - x^{*}(x)| \leq |x_n^{*}(x_n) - x_n^{*}(x)|+|x_n^{*}(x) - x^{*}(x)|$$ $$ \leq \|x_n^{*}\| ||x_n-x\|+|x_n^{*}(x) - x_n^{*}(x)|.$$ The second term tends to $0$ by weak* convergence of $(x_n^{*})$ and the first term tends to $0$ because $\|x_n^{*}\|$ is bounded.
For a counter-example when we only have weak convergence of $(x_n)$ consider a Hilbert space with orthonormal basis $(e_n)$. Take $x_n=x_n^{*}=e_n$ and $x=x^{*}=0$.
2) is proved similarly.