Closed graph theorem
Consider the product of two Banach spaces, , - which is just the linear space of all pairs and , is a Banach space with respect to the sum of the norms
Theorem (Closed Graph Theorem)
If is a linear map between Banach spaces then it is bounded if and only if its graphis a closed subset of the Banach space .
If is a linear map between Banach spaces then it is bounded if and only if its graphis a closed subset of the Banach space .
Proof of Closed Graph Theorem
Proof of Closed Graph Theorem
Conversely, suppose the graph is closed. This means that viewed as a normed space in its own right it is complete. Given the graph we can reconstruct the map it comes from (whether linear or not) in a little diagram. From consider the two projections, and . Both of them are continuous since the norm of either or is less than the norm in (1).Indeed there are two ways to map a point to , either directly, sending it to or first sending it to and then to . Since these are the same.
Now, as already noted, is a closed subspace, so it too is a Banach space and and remain continuous when restricted to it. The map is 1-1 and onto, because each occurs as the first element of precisely one pair, namely . Thus the Corollary above applies to to show that its inverse, is continuous. But then , from the commutativity, is also continuous proving the theorem.