# proof that ${L}^{p}$ spaces are complete

Let’s prove completeness for the classical Banach spaces, say ${L}^{p}[0,1]$ where $p\ge 1$.

Since the case $p=\mathrm{\infty}$ is elementary, we may assume $$.
Let $[{f}_{\cdot}]\in {({L}^{p})}^{\mathbf{N}}$ be a Cauchy sequence^{}. Define $[{g}_{0}]:=[{f}_{0}]$ and for $n>0$ define $[{g}_{n}]:=[{f}_{n}-{f}_{n-1}]$. Then $[{\sum}_{n=0}^{N}{g}_{n}]=[{f}_{N}]$ and we see that

$$ |

Thus it suffices to prove that etc.

It suffices to prove that each absolutely summable series in ${L}^{p}$ is summable in ${L}^{p}$ to some element in ${L}^{p}$.

Let $\{{f}_{n}\}$ be a sequence^{} in ${L}^{p}$ with
$$, and define functions ${g}_{n}$ by
setting ${g}_{n}(x)={\sum}_{k=1}^{n}|{f}_{k}(x)|$. From the Minkowski inequality^{} we
have

$$\parallel {g}_{n}\parallel \le \sum _{k=1}^{n}\parallel {f}_{k}\parallel \le M.$$ |

Hence

$$\int {g}_{n}^{p}\le {M}^{p}.$$ |

For each $x$, $\{{g}_{n}(x)\}$ is an increasing sequence of (extended) real
numbers and so must converge^{} to an extended real number $g(x)$. The
function $g$ so defined is measurable, and, since ${g}_{n}\ge 0$, we have

$$\int {g}^{p}\le {M}^{p}$$ |

by Fatou’s Lemma. Hence ${g}^{p}$ is integrable, and $g(x)$ is finite for almost all $x$.

For each $x$ such that $g(x)$ is finite the series ${\sum}_{k=1}^{\mathrm{\infty}}{f}_{k}(x)$ is an absolutely summable series of real numbers and so must be summable to a real number $s(x)$. If we set $s(x)=0$ for those $x$ where $g(x)=\mathrm{\infty}$, we have defined a function $s$ which is the limit almost everywhere of the partial sums ${s}_{n}={\sum}_{k=1}^{n}{f}_{k}$. Hence $s$ is measurable. Since $|{s}_{n}(x)|\le g(x)$, we have $|s(x)|\le g(x)$. Consequently, $s$ is in ${L}^{p}$ and we have

$${|{s}_{n}(x)-s(x)|}^{p}\le {2}^{p}{[g(x)]}^{p}.$$ |

Since ${2}^{p}{g}^{p}$ is integrable and ${|{s}_{n}(x)-s(x)|}^{p}$ converges to $0$ for almost all $x$, we have

$$\int {|{s}_{n}-s|}^{p}\to 0$$ |

by the Lebesgue Convergence Theorem. Thus ${\parallel {s}_{n}-s\parallel}^{p}\to 0$, whence $\parallel {s}_{n}-s\parallel \to 0$. Consequently, the series $\{{f}_{n}\}$ has in ${L}^{p}$ the sum $s$.

## References

Royden, H. L. *Real analysis. Third edition*. Macmillan Publishing Company, New York, 1988.

Title | proof that ${L}^{p}$ spaces are complete^{} |
---|---|

Canonical name | ProofThatLpSpacesAreComplete |

Date of creation | 2013-03-22 14:40:09 |

Last modified on | 2013-03-22 14:40:09 |

Owner | Simone (5904) |

Last modified by | Simone (5904) |

Numerical id | 8 |

Author | Simone (5904) |

Entry type | Proof |

Classification | msc 46B25 |