# proof of necessary and sufficient conditions for a normed vector space to be a Banach space

We prove here that in order for a normed space^{}, say $X$, with the norm, say $\parallel \cdot \parallel $ to be
Banach, it is necessary and sufficient that convergence of every absolutely convergent series in $X$
implies convergence of the series in $X$.

Suppose that $X$ is Banach. Let a sequence^{} $({x}_{n})$ be in $X$ such that the series

$$\sum _{n}\parallel {x}_{n}\parallel $$ |

converges. Then for all $\u03f5>0$ there exists $N$ such that for all $m>n>N$ we have

$$ |

Hence

$${s}_{k}=\sum _{n=1}^{k}{x}_{n}$$ |

is a Cauchy sequence^{} in $X$. Since $X$ is Banach, ${s}_{k}$ converges in $X$.

Conversely, suppose that absolute convergence^{} implies convergence. Let $({x}_{n})$ be a Cauchy sequence
in $X$. Then for all $m\ge 1$ there exists ${N}_{m}$ such that for all $k,{k}^{\prime}\ge {N}_{m}$ we have $$.
We’ll conveniently choose ${N}_{m}$ so that ${N}_{m}$ is an increasing sequence in $m$. Then in particular, $$. Hence we have,

$$ |

The sum on the right converges, so must the sum on the left. Since absolute convergence implies convergence, we must have

$$\sum _{m=1}^{M}({x}_{{N}_{m}}-{x}_{{N}_{m+1}})$$ |

converges as $M$ tends to infinity. So there is an $s$ in $X$ which is the limit of the sum above. As a telescoping
series, however, the sum above converges to ${lim}_{M\to \mathrm{\infty}}({x}_{{N}_{1}}-{x}_{{N}_{m}})=s$. Since $s$ and ${x}_{{N}_{1}}$ are both in $X$, so is the limit of ${x}_{{N}_{m}}$, which is a subsequence of the Cauchy sequence $({x}_{n})$. Hence $({x}_{n})$ converges in $X$. So $X$ is Banach.

This completes^{} the proof.

Title | proof of necessary and sufficient conditions for a normed vector space to be a Banach space^{} |
---|---|

Canonical name | ProofOfNecessaryAndSufficientConditionsForANormedVectorSpaceToBeABanachSpace |

Date of creation | 2013-03-22 17:35:11 |

Last modified on | 2013-03-22 17:35:11 |

Owner | willny (13192) |

Last modified by | willny (13192) |

Numerical id | 4 |

Author | willny (13192) |

Entry type | Proof |

Classification | msc 46B99 |