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 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:
Noting that
is an element of (by construction of )
and that and were arbitrary,
we conclude that every neighborhood of contains an element of ,
for all in .
This proves that is dense in 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 |