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The Riesz representation theorem allows identifying elements of the dual space with elements of the Hilbert space $H$ itself via the inner product. This adapts the general definition of weak convergence (which uses dual space functions) to the inner product form used specifically in Hilbert spaces.

Conditions: $H$ is a Hilbert space.; The general definition of weak convergence involves continuous linear functionals from the dual space.

The Riesz representation theorem allows identifying elements of the dual space with elements of the Hilbert space $H$ itself via the inner product. This adapts the general definition of weak convergence (which uses dual space functions) to the inner product form used specifically in Hilbert spaces.

Conditions: $H$ is a Hilbert space.; The general definition of weak convergence involves continuous linear functionals from the dual space.

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 $H$.