quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm
Let be an absolutely convergent series in , i.e., . By definition of the quotient norm, there exists such that
It is clear that and so, as is a Banach space, is convergent.
Let and . We have that
Since we see that converges in to .
|Title||quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm|
|Date of creation||2013-03-22 17:23:01|
|Last modified on||2013-03-22 17:23:01|
|Last modified by||asteroid (17536)|