I have just learned weak convergence, that is defined as follows:
Definition Let $X$ be a Banach space, $\{x_n\}\subset X$ be a sequence, and $x\in X$. $\{x_n\}$ is said to be weakly convergent to $x$ if $f(x_n) \to f(x)$ for any $f\in X^*$.
And we use the notation $x_n \buildrel{w}\over{\to} x$ for weak convergence.
What I was wondering is that, is there a commonly accepted notation for the weak limit that corresponds to $\displaystyle\lim_{n\to \infty}$? I thought it would be convenient to have one, but my textbook or any information on the internet does not talk about this issue. Is there a mathematical reason for why we don't have one (if there isn't), or is it just tradition?
As mentioned in the comments, the notation $$w-\lim_{n \to \infty}x_n =x$$ is used in certain places. One reference I could quickly find for this was Functional Analysis by K. Yosida.