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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