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.