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No, weak convergence alone does not imply strong convergence. A sequence can converge weakly to a limit without the norm of the difference converging to zero.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging weakly to $l$.; No assumption is made about the convergence of $\|x_n\|$ to $\|l\|$.

No, weak convergence alone does not imply strong convergence. A sequence can converge weakly to a limit without the norm of the difference converging to zero.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$ converging weakly to $l$.; No assumption is made about the convergence of $\|x_n\|$ to $\|l\|$.