What is the definition of strong convergence in a Hilbert space?

Conditions

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

Reasoning, step by step

1. Identify the sequence $(x_n)$ and the candidate limit $l$ in the Hilbert space $H$.
2. Calculate the difference between the sequence terms and the limit: $x_n - l$.
3. Take the norm of this difference: $\|x_n - l\|$.
4. Evaluate the limit of the norm as $n$ approaches infinity.
5. Conclude that the sequence strongly converges to $l$ if $\lim_{n \to \infty} \|x_n - l\| = 0$.

Example

The slide explicitly states: 'A sequence ($x_n$) is strongly convergent to l if $\lim _{n\to \infty }$ || $x_n - l$ || = 0'.

Common misconceptions

• Confusing strong convergence with weak convergence, which uses inner products instead of norms.
• Assuming that the sequence terms themselves must approach zero, rather than the distance to the limit $l$.

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Explore next

• How is weak convergence defined using inner products in a Hilbert space?
• Why does strong convergence imply weak convergence?