proof of closed graph theorem
Let be a linear mapping. Denote its graph by , and let and be the projections onto and , respectively. We remark that these projections are continuous, by definition of the product of Banach spaces.
If is bounded, then given a sequence in which converges to , we have that
and
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 |
---|---|
Canonical name | ProofOfClosedGraphTheorem |
Date of creation | 2013-03-22 14:48:47 |
Last modified on | 2013-03-22 14:48:47 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 5 |
Author | Koro (127) |
Entry type | Proof |
Classification | msc 46A30 |