Thursday, March 30. 2006
Separability of Banach spaces
I mulled repeatedly for some weeks over the fact, why a Banach space is separable if any element can be approximated by a sequence of finite dimensional elements. It has been formulated as , where
is a continuous linear operator with finite rank, and this conclusion was never obvious to me. For another time, there was a fact which should be clear and not difficult to show.
Finally, I remembered: Every (finite dimensional) vector space has a (finite) basis. I.e., if , then there is a basis
which can be extended as soon as
for any
. Any basis is countable, and a theorem says that a space
is separable iff there is a countable set
with
, where
is the closure of the set of all (finite) linear combinations of elements in
.
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