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A sequence $(x_n)$ in a Hilbert space $H$ is weakly convergent to $l$ if for every element $v$ in $H$, the inner product of $x_n$ with $v$ converges to the inner product of $l$ with $v$ as $n$ approaches infinity. This is denoted by $x_n \rightharpoonup l$.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $l$ is an element of $H$.; $v$ is an arbitrary element of $H$.

A sequence $(x_n)$ in a Hilbert space $H$ is weakly convergent to $l$ if for every element $v$ in $H$, the inner product of $x_n$ with $v$ converges to the inner product of $l$ with $v$ as $n$ approaches infinity. This is denoted by $x_n \rightharpoonup l$.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a sequence in $H$.; $l$ is an element of $H$.; $v$ is an arbitrary element of $H$.