Weak * convergent vs strongly convergent. I read in Kreyszig the following definitions.
$\textbf{Definition:}$ Let $X,Y$ normed spaces and $T_n:X \rightarrow Y$ a sequence of bounded linear operators. We say that $T_n$ converges strongly to $T:X\rightarrow Y$ if
$$ \Vert T_n(x)- T(x) \Vert \to 0 , \forall x\in X $$
$\textbf{Definition:}$ Let $X$ a normed space and $f_n \in X'$ a sequence. We say that $f_n$ converges weakly * if exists $f\in X'$ such that
$$ \vert f_n(x) - f(x) \vert \to 0, \forall x\in X $$
In the first case, $T$ may be unbounded if $X$ is not complete. Put $Y=\mathbb{R}$ the definitions are almost the same except for the fact that in the second they tell that $f$ has to be continuous and in the first not.
My doubt comes because the author comments that in the case of working with linear functionals the first definition coincides with the second, but I think they are not the same.
See pag. 266 of Kreyszig: Introductory Functional Analysis with applications.

The two definitions are not equivalent when $Y=\mathbb C$ and $X$ is not complete.  Let $X$ be the space of trigonometric polynomials on $[0,2\pi]$ with the $L^2$-norm, and let $f_n:X\to\mathbb C$ be defined by linear extension of
$$f_n(e^{ikx})=\left\{\begin{array}{ll}
1 & \text{if }|k|\leq n,\\
0 & \text{otherwise.}
\end{array}\right.$$
Then the sequence $\{f_n\}$ in $X^*$ converges strongly to an unbounded functional, hence cannot converge weak$^*$ to a bounded functional.
