Every infinite-dimensional banach space permits a strictly descending chain of dense linear subspaces.
The proof demonstrates that a strictly ascending chain of dense linear subspaces exists as well.
These chains can achieve a length corresponding to the space's cardinality.
The findings enhance understanding of the structure of linear subspaces in banach spaces.
Resumen
It is proved that every infinite-dimensional Banach space X of cardinality m admits both a strictly descending chain and a strictly ascending chain of dense linear subspaces of length m.