On basic sequences in Banach spaces
Abstract
Let X be a Banach space with
separable. If X has a shrinking basis and Y is a closed subspace of
which contains X, there exists a shrinking basis
in X with two complementary subsequences
and
so that
is a reflexive space and
, where we are denoting by
the weak-star closure of
in
. If
is a sequence in X that converges to a point in
for the weak-star topology,there is a basic sequence
in
such that
is a quasi-reflexive Banach space of order one. Given a Banach space Z with basis it is also proved that every basic sequence
in Z has a subsequence extending to a basis of Z.
![X<sup>**</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e3070a4f19e48e449273bb3ddc332541.png)
![X<sup>**</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e3070a4f19e48e449273bb3ddc332541.png)
![(x<sub>n</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/151d68e990b62240ea8ae3b13f1fe87a.png)
![(x_{m<sub>i</sub>})](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e71a439555cc0f5dc36c305828e2e662.png)
![(x_{n<sub>j</sub>})](http://siba-ese.unile.it/plugins/generic/latexRender/cache/5f18c7c2f306d460238185909d928623.png)
![[x_{m<sub>j</sub>}]](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f1ce0d91842a11df6d8ddebf158be264.png)
![X +[\widetilde{x_{n<sub>j</sub>}}]= Y](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b5bd32e4e3f113ffcd56a0a047d15551.png)
![[\widetilde{x_{n<sub>j</sub>}}]](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f44530db2352e12c1f8e9bbda57cfbc1.png)
![[x_{n<sub>j</sub>}]](http://siba-ese.unile.it/plugins/generic/latexRender/cache/2599d2644c7e374f98aa7b6e7d716aa4.png)
![X<sup>**</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e3070a4f19e48e449273bb3ddc332541.png)
![(y<sub>n</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/7779fed5f88c313b7930dad791ef378e.png)
![X<sup>**</sup>\thicksim X](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8d96a64e84e44ecb91bd67bd0e5f7642.png)
![( y_{n<sub>j</sub>})](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6b9ea9400f4ba7e1c5bda6e0758ed64c.png)
![(y<sub>n</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/7779fed5f88c313b7930dad791ef378e.png)
![[y_{n<sub>j</sub>}]](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1ea11f01e20b017e70d9904d880c136b.png)
![(z<sub>n</sub>)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/4e33817d586bbaf69768e128806a6d30.png)
DOI Code:
10.1285/i15900932v12p245
Full Text: PDF