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The extra assumption required is that the norms of the sequence converge to the norm of the limit, i.e., $\|x_n\| \to \|l\|$. When combined with weak convergence, this condition implies strong convergence.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging weakly to $l$.; $\|x_n\|$ converges to $\|l\|$.

The extra assumption required is that the norms of the sequence converge to the norm of the limit, i.e., $\|x_n\| \to \|l\|$. When combined with weak convergence, this condition implies strong convergence.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging weakly to $l$.; $\|x_n\|$ converges to $\|l\|$.