Exercise · ZW-RIESZ-FISCHER-THEOREM-006 · ★★☆ Intermediate
If \( \|f_{n+1}-f_n\|_p<2^{-n} \) for all \( n\ge 0 \), what bound is obtained for \( \sum_{n\ge 0} \|f_{n+1}-f_n\|_p \)?
Choose one option.
Show answer
- D. It is at most \( 2 \)
Explanation
With \( n\ge 0 \) the geometric sum \( \sum_{n\ge 0} 2^{-n} \) equals \( 2 \), so the bound is \( 2 \). The other options do not follow from the hypothesis.
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-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) ?
- ZW-RIESZ-FISCHER-THEOREM-009 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→∞ ?
← ZW-RIESZ-FISCHER-THEOREM-005 ZW-RIESZ-FISCHER-THEOREM-007 →