A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other The functionals
$$
  \phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t
$$
define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$.
a) Show that $(\phi_n)$ converges *-weakly in $C^1([-1,1])'$.
b) Does $(\phi_n)$ converges *-weakly in $C([-1,1])'$?
For me the limit functional
$$
 \int_{0 \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t
$$
is not well defined so i have trouble evaluating the condition of convergence? Do you have any hints?
 A: We can write 
$$\phi_n(x)=\int_{1/n}^1\frac{x(t)-x(-t)}tdt.$$
When $x$ is in $C^1[-1,1]$, this converge to $\int_0^1\frac{x(t)-x(-t)}tdt$ (and this integral is convergent, as the problem at $0$ is solved by the derivative. 
Taking a continuous function $f$ such that $f=1$ on $[n^{-1},1]$ and $-1$ on $[-1,-n^{-1}]$, we can see that $\lVert \phi_n\rVert_{(C[-1,1])'}=2\log n$ so we can't have weak$^*$ convergence in $(C[-1,1])'$.
A: @Davide Giraduo
Regarding your derivation, i came to another result:
\begin{align*}
 \int_{1/n \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t & = \int_{-1}^{-1/n} \frac{x(t)}{t} \mathrm{d} t +  \int_{1/n}^1 \frac{x(t)}{t} \mathrm{d} t \\ 
                                                                                 & =  - \int_{1}^{1/n} \frac{x(-t)}{t} \mathrm{d} t +  \int_{1/n}^1 \frac{x(t)}{t} \mathrm{d} t \\ 
   & =  \int_{1/n}^{1} \frac{x(-t)}{t} \mathrm{d} t +  \int_{1/n}^1 \frac{x(t)}{t} \mathrm{d} t \\
                      & = \int_{1/n}^{1} \frac{x(-t) + x(t)}{t} \, \mathrm{d} t
\end{align*}
Which has others sign's?
