# every normed space with Schauder basis is separable

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 $\mathbb{C}$. So let $(X,\left\|\cdot\right\|)$ be a normed space with Schauder basis, say $S=\left\{e_{1},e_{2},\dots\right\}$. 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 $q_{1}e_{1}+\cdots+q_{n}e_{n}$ 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 $\epsilon>0$. 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 $n\geq N$ we have,

 $\left\|\sum_{j=1}^{n}\alpha_{j}e_{j}-x\right\|<\epsilon/2$

But then in particular,

 $\left\|\sum_{j=1}^{N}\alpha_{j}e_{j}-x\right\|<\epsilon/2$

Furthermore, by density of $\mathbb{Q}$ in $\mathbb{R}$, we know that there exist constants $a_{1},\dots,a_{N},b_{1},\dots,b_{N}$ in $\mathbb{Q}$ such that,

 $\left\|\sum_{j=1}^{N}(a_{j}+b_{j}i)e_{j}-\sum_{j=1}^{N}\alpha_{j}e_{j}\right\|% <\epsilon/2$

By triangle inequality we obtain:

 $\left\|\sum_{j=1}^{N}(a_{j}+b_{j}i)e_{j}-x\right\|\leq\left\|\sum_{j=1}^{N}(a_% {j}+b_{j}i)e_{j}-\sum_{j=1}^{N}\alpha_{j}e_{j}\right\|+\left\|\sum_{j=1}^{N}% \alpha_{j}e_{j}-x\right\|<\epsilon$

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 $\epsilon$ 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 EveryNormedSpaceWithSchauderBasisIsSeparable 2013-03-22 17:36:07 2013-03-22 17:36:07 asteroid (17536) asteroid (17536) 11 asteroid (17536) Theorem msc 42-00 msc 15A03 msc 46B15