every normed space with Schauder basis is separable
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 where is either or . So let be a normed space with Schauder basis, say . Notice that our notation implies that is infinite. In finite dimensional case, the same proof with a slight modification will yield the result.
Now, set to be the set of all finite sums such that each where . Clearly is countable. It remains to show that is dense (http://planetmath.org/Dense) in .
Let . Let . By definition of Schauder basis, there is a sequence of scalars and there exists such that for all we have,
But then in particular,
Furthermore, by density of in , we know that there exist constants in such that,
By triangle inequality we obtain:
|Title||every normed space with Schauder basis is separable|
|Date of creation||2013-03-22 17:36:07|
|Last modified on||2013-03-22 17:36:07|
|Last modified by||asteroid (17536)|