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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 $\sup_n \|x_n\|_H < \infty$.

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 $\|l\|$.