Example of weakly convergent sequence, not convergent in norm

Question

I need to find example of sequence $(f_n)_{n=1}^{\infty}$ in $C[a,b]$ which converges weakly to $f \in C[a,b]$, such that $||f_n||_u \to ||f||_u, n \to \infty$, where $$||f||_u = \underset{x \in [a,b]}{max }|f(x)|,$$ but $(f_n)_{n=1}^{\infty}$ doesn't converge to $f$ in norm.

Can you help me, please?

Answer 1

Say $[a,b] = [0,1]$ and take for $f_n$ function which is:
a. 1 at 0
b. 0 at $\frac{1}{2n}$
c. 1 at $\frac{1}{n}$
d. 1 at 1
e. and linear between these points
The weak limit of $f_n$ is function which is identically $1$ since we have point-wise convergence and functions are bounded. At the same time the norm of the difference is $1$.