A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is an $x\in X$ such that $\lim_{n\rightarrow \infty } | x_n - x| =0 $
Compare weak convergence : A sequence $(x_n)$ in a normed space $X$ is said to be weakly convergent if there is an $x\in X$ such that for every $f\in X'$, $\lim_{n\rightarrow \infty } f( x_n ) = f(x) $ where $X'$ is a dual space. In fact if ${\rm dim}\ X<\infty$ they are same. In general weak convergene does not imply strong convergence
