What does the Kakutani theorem say about bounded sequences in a Hilbert space?

Conditions

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

Reasoning, step by step

1. Identify a sequence $(x_n)$ in $H$.
2. Verify that the sequence is bounded, i.e., the norms $\|x_n\|_H$ are bounded.
3. Apply the Kakutani theorem.
4. Conclude that there exists a subsequence of $(x_n)$ that converges weakly in $H$.

Example

The compactness slide states: 'Let ($x_n$) be a bounded sequence in H. (i.e. the norm ||$x_n$||_H is bounded)... Thanks to the Kakutani Theorem, one can extract a subsequence weakly converging in H.'

Common misconceptions

• Assuming that the entire sequence converges weakly, rather than just a subsequence.
• Thinking that boundedness implies strong convergence, which is false in infinite-dimensional spaces.

Watch the explanation

Explore next

• What extra assumption turns weak convergence into strong convergence?
• How is weak convergence defined using inner products in a Hilbert space?