Exercise · ZW-RIESZ-FISCHER-THEOREM-009 · ★★☆ Intermediate
Let \( (f_n) \) be a Cauchy sequence in \( L^2 \) with limit \( f \) in \( L^2 \). What happens to \( \|f_n-f\|_2 \) as \( n\to\infty \)?
Choose one option.
Show answer
- A. It tends to zero
Explanation
Convergence in norm means the distance to the limit becomes small. The other options contradict completeness.
Review the concept: Riesz–Fischer theorem
The Riesz-Fischer theorem states that the space \( L^p(\mu) \) is complete for \( 1 \le p \le \infty \), that is, every Cauchy sequence in it converges to an element of \( L^p(\mu) \).
\[\lim_{n,m\to\infty}\|f_n-f_m\|_p=0 \implies \exists f\in L^p:\ \|f_n-f\|_p\to 0\]
More practice problems: riesz–fischer theorem
- ZW-RIESZ-FISCHER-THEOREM-001 Which space does the Riesz-Fischer theorem guarantee is complete?
- ZW-RIESZ-FISCHER-THEOREM-002 What type of sequence appears as the hypothesis of the theorem?
- ZW-RIESZ-FISCHER-THEOREM-003 Which inequality is used to bound the norm of the sum of differences?
- ZW-RIESZ-FISCHER-THEOREM-004 What type of convergence is obtained for the constructed subsequence?
- ZW-RIESZ-FISCHER-THEOREM-005 In which space is the theorem verified for p=2 ?
- ZW-RIESZ-FISCHER-THEOREM-006 If \|f_n+1-f_n\|_p<2^-n for all n≥ 0 , what bound is obtained for ∑_n≥ 0 \|f_n+1-f_n\|_p ?
- ZW-RIESZ-FISCHER-THEOREM-007 If g=∑_k |h_k| with \|g\|_p≤ 1 , what follows about g ?
- ZW-RIESZ-FISCHER-THEOREM-008 What is the limit in L^1([0,1]) of f_n=1_(0,1/n) ?
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