Convergence of linear map on a subset implies convergence on the span? Suppose that $X$ is a Banach space, and that I have a sequence of linear maps $(T_{n})_{n\geq1}$ such that $T_{n}(x) \to T(x)$ for all $x\in K$ where, $K \subset X$. 
Since $T$ is linear does it follow that $T_{n}(x) \to T(x)$ for all $x\in \rm span(K)$? In finite dimensions this is clear, but I am not sure if anything changes in infinite dimensions since I am unclear on the definition of $\rm span(K)$ in this case (my only knowledge of $\rm span$ comes from linear algebra, so I am not sure if an alternate and more general definition is required).

If the definition is unchanged in the infinite case as suggested by Alex R., then the answer is trivial since if $w \in \rm span(K)$ then $w$ can be written as $w= \sum_{j=1}^{J(w)} \lambda_{j}w_{j}$ where $w_{j} \in K$ for all $j$, and $J(w)<\infty$ is a finite number which depends on $w$. Thus: $$T_{n}(w) = \sum_{j=1}^{J} \lambda_{j}T_{n}(w_{j}) \rightarrow \sum_{j=1}^{J} \lambda_{j}T(w_{j}) = T(w)$$
