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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 strongly convergent to $l$ if the norm of the difference between $x_n$ and $l$ approaches zero as $n$ approaches infinity. This is denoted by $x_n \rightarrow l$.

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

A sequence $(x_n)$ in a Hilbert space $H$ is strongly convergent to $l$ if the norm of the difference between $x_n$ and $l$ approaches zero as $n$ approaches infinity. This is denoted by $x_n \rightarrow l$.

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

The Kakutani theorem states that every bounded sequence in a Hilbert space $H$ admits a subsequence that converges weakly in $H$. This provides a compactness principle for weak convergence in infinite-dimensional spaces.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a bounded sequence in $H$, meaning $\sup_n \|x_n\|_H < \infty$.

The Kakutani theorem states that every bounded sequence in a Hilbert space $H$ admits a subsequence that converges weakly in $H$. This provides a compactness principle for weak convergence in infinite-dimensional spaces.

Conditions: $H$ is a Hilbert space.; $(x_n)$ is a bounded sequence in $H$, meaning $\sup_n \|x_n\|_H < \infty$.