Given the sequence of functionals $$f_n(x)=\int_{\frac{1}{n}\leq|t|\leq1}\frac{x(t)}{t}dt$$ in $C([-1,1])$ respectively $C^1([-1,1])$.
How can I show that $(f_n)$ weak-*-converges in $C^1([-1,1])^*$?
And answer the question if it does in $C([-1,1])^*$.
1) Intuitively, the weak-star limit should be the functional $$ u \mapsto \int_{[-1,1]} t^{-1} u(t) dt. $$ Then show that $$ \int_{-1/n}^{1/n} t^{-1} u(t) dt $$ tends to zero for fixed $u$ by partial integration.
2) Show that these functionals are unbounded in $C([-1,1])^*$: For each $n$, take a piecewise linear function, which is equal to one on $[1/n,1]$ and zero on $[-1,0]$.