# every normed space with Schauder basis is separable

\PMlinkescapephrase

dense in

Here we show that every normed space that has a Schauder basis is
separable^{} (http://planetmath.org/Separable).
Note that we are (implicitly) assuming that the normed spaces in question
are spaces over the field $K$ where $K$ is either $\mathbb{R}$ or $\u2102$.
So let $(X,\parallel \cdot \parallel )$ be a normed space with Schauder basis,
say $S=\{{e}_{1},{e}_{2},\mathrm{\dots}\}$.
Notice that our notation implies that $S$ is infinite.
In finite dimensional case,
the same proof with a slight modification will yield the result.

Now, set $Q$ to be the set of all finite sums ${q}_{1}{e}_{1}+\mathrm{\cdots}+{q}_{n}{e}_{n}$ such that each ${q}_{j}={a}_{j}+{b}_{j}i$ where ${a}_{j},{b}_{j}\in \mathbb{Q}$. Clearly $Q$ is countable. It remains to show that $Q$ is dense (http://planetmath.org/Dense) in $X$.

Let $\u03f5>0$. Let $x\in X$. By definition of Schauder basis, there is a sequence of scalars $({\alpha}_{n})$ and there exists $N$ such that for all $n\ge N$ we have,

$$ |

But then in particular,

$$ |

Furthermore, by density of $\mathbb{Q}$ in $\mathbb{R}$, we know that there exist constants ${a}_{1},\mathrm{\dots},{a}_{N},{b}_{1},\mathrm{\dots},{b}_{N}$ in $\mathbb{Q}$ such that,

$$ |

By triangle inequality^{} we obtain:

$$ |

Noting that

$$\sum _{j=1}^{N}({a}_{j}+{b}_{j}i){e}_{j}$$ |

is an element of $Q$ (by construction of $Q$)
and that $x$ and $\u03f5$ were arbitrary,
we conclude that every neighborhood^{} of $x$ contains an element of $Q$,
for all $x$ in $X$.
This proves that $Q$ is dense in $X$ and completes^{} the proof.

Title | every normed space with Schauder basis is separable |
---|---|

Canonical name | EveryNormedSpaceWithSchauderBasisIsSeparable |

Date of creation | 2013-03-22 17:36:07 |

Last modified on | 2013-03-22 17:36:07 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 11 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 42-00 |

Classification | msc 15A03 |

Classification | msc 46B15 |