Let $X$ be a topological vector space.

Let us say that $X$ enjoys *sequential separablity* if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{x_n\}$ converging to $x$

Example. Suppose that $X$ is a TVS with $X=\bigcup_1^{\infty} X_n$ such that all $X_n$'s are relatively separable metrizable. Then $X$ is sequentially separable. Thus, every seprable normed space is sequential separable. Morover the weak-star topology on the dual of separable normed spaces is also sequntialy separable.

Q. Any example of a separable TVS which is not sequentially separable?

Indeed, for a given separable TVS, what are the necessary or sufficient conditions for sequentially separable property?

approximation propertyis a well established important notion for Banach (and topological vector) spaces. You should look for a different name. $\endgroup$sequentially separable? $\endgroup$issequentially separable, or elseenjoyssequential separability. $\endgroup$There exists a sequence $(x_n)_n$ in $X$ such that for any $x\in X$ there exists a subsequence of $(x_n)$ converging to $x$.$\endgroup$1more comment