Lp spaces · Definition

Riesz–Fischer theorem

measure theory and integrationlp spaces

Before you start you need to know

The \( L^p \) spaces are the natural setting of modern functional analysis. This theorem guarantees that Cauchy sequences do not escape from them.

Definition

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 \( L^p(\mu) \).

\[\lim_{n,m\to\infty}\|f_n-f_m\|_p=0 \implies \exists f\in L^p:\ \|f_n-f\|_p\to 0\]

If \( (f_n) \) is Cauchy in \( L^p(\mu) \), one extracts a subsequence \( (f_{n_k}) \) with \( \|f_{n_{k+1}}-f_{n_k}\|_p < 2^{-k} \). By Minkowski's inequality, the function \( g=\sum_k |f_{n_{k+1}}-f_{n_k}| \) satisfies \( \|g\|_p \le 1 \), so it is finite almost everywhere. The series \( f_{n_1}+\sum_k (f_{n_{k+1}}-f_{n_k}) \) converges absolutely a.e. to a limit \( f \).

Take \( \mu \) to be Lebesgue measure on \([0,1]\) and \( f_n = n\,\mathbf{1}_{(0,1/n)} \) in \( L^1 \). Since \( \|f_n-f_m\|_1 \) does not tend to zero, this sequence is not Cauchy; instead \( f_n=\mathbf{1}_{(0,1/n)} \) in \( L^1 \) is Cauchy and converges to \( 0 \) in norm. Fatou's lemma then verifies that \( \|f_n-f\|_p \to 0 \).

A typical error is to conclude that pointwise convergence of the subsequence implies convergence in \( L^p \) without proving the bound on \( g \). Without Minkowski's inequality there is no control over \( g \), and the argument collapses.

Think about it

Why is \( C[0,1] \) with the \( L^1 \) norm not a complete space?

Because a sequence of continuous functions can converge in \( L^1 \) to a discontinuous function, such as an indicator, which is not in \( C[0,1] \).

Exercises 10

  1. Which space does the Riesz-Fischer theorem guarantee is complete?
  2. What type of sequence appears as the hypothesis of the theorem?
  3. Which inequality is used to bound the norm of the sum of differences?
  4. What type of convergence is obtained for the constructed subsequence?
  5. In which space is the theorem verified for \( p=2 \)?
  6. If \( \|f_{n+1}-f_n\|_p<2^{-n} \) for all \( n\ge 0 \), what bound is obtained for \( \sum_{n\ge 0} \|f_{n+1}-f_n\|_p \)?
  7. If \( g=\sum_k |h_k| \) with \( \|g\|_p\le 1 \), what follows about \( g \)?
  8. What is the limit in \( L^1([0,1]) \) of \( f_n=\mathbf{1}_{(0,1/n)} \)?
  9. Let \( (f_n) \) be a Cauchy sequence in \( L^2 \) with limit \( f \) in \( L^2 \). What happens to \( \|f_n-f\|_2 \) as \( n\to\infty \)?
  10. Let \( f_n=\mathbf{1}_{(n,n+1)} \) in \( L^1(\mathbb{R}) \). Is it Cauchy? If so, does it have a limit in \( L^1 \)?

References

  • Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edition, Wiley, 1999.
  • H. L. Royden, P. M. Fitzpatrick, Real Analysis, 4th edition, Pearson, 2010.