Does weak convergence imply strong convergence?

Conditions

• $H$ is a Hilbert space.
• $(x_n)$ is a sequence in $H$ converging weakly to $l$.
• No assumption is made about the convergence of $\|x_n\|$ to $\|l\|$.

Reasoning, step by step

1. Recall the definition of weak convergence: $\langle x_n, v \rangle \to \langle l, v \rangle$ for all $v \in H$.
2. Recall the definition of strong convergence: $\|x_n - l\| \to 0$.
3. Observe that weak convergence controls the behavior of the sequence against all test vectors, but does not directly control the distance $\|x_n - l\|$.
4. Conclude that weak convergence does not automatically imply strong convergence.

Example

The lecturer explicitly states that a weakly convergent sequence is not necessarily strongly convergent. The proposition slide adds the extra assumption that $\|x_n\|$ converges to $\|l\|$ before concluding strong convergence.

Common misconceptions

• Assuming that because strong convergence implies weak convergence, the converse must also be true.
• Thinking that weak convergence is a stronger condition than strong convergence.

Watch the explanation

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