proof of closed graph theorem
by continuity of the projections. But then, since is continuous,
Thus , proving that is closed.
Now suppose is closed. We remark that is a vector subspace of , and being closed, it is a Banach space. Consider the operator defined by . It is clear that is a bijection, its inverse being , the restriction of to . Since is continuous on , the restriction is continuous as well; and since it is also surjective, the open mapping theorem implies that is an open mapping, so its inverse must be continuous. That is, is continuous, and consequently is continuous.
|Title||proof of closed graph theorem|
|Date of creation||2013-03-22 14:48:47|
|Last modified on||2013-03-22 14:48:47|
|Last modified by||Koro (127)|