Definition 23.1.11 Let f : D → E. The set of all ordered pairs of the form {(x,f(x)) : x ∈ D} is called the graph of f.
Definition 23.1.12 If X and Y are normed linear spaces, make X × Y into a normed linear space by using the norm (x,y) = max
There are other ways to give a norm for X × Y . For example, you could define (x,y) = x + y
Lemma 23.1.13 The norm defined in Definition 23.1.12 on X ×Y along with the definition of addition and scalar multiplication given there make X × Y into a normed linear space.
Proof: The only axiom for a norm which is not obvious is the triangle inequality. Therefore, consider
Lemma 23.1.14 If X and Y are Banach spaces, then X × Y with the norm and vector space operations defined in Definition 23.1.12 is also a Banach space.
Proof: The only thing left to check is that the space is complete. But this follows from the simple observation that
Proof: If F ⊆ X where X is a Banach space and
Definition 23.1.16 Let X and Y be Banach spaces and let D ⊆ X be a subspace. A linear map L : D → Y is said to be closed if its graph is a closed subspace of X ×Y . Equivalently, L is closed if x_{n} → x and Lx_{n} → y implies x ∈ D and y = Lx.
Note the distinction between closed and continuous. If the operator is closed the assertion that y = Lx only follows if it is known that the sequence

where the norm is the uniform norm on C

Therefore, if

and so by the fundamental theorem of calculus f
The next theorem, the closed graph theorem, gives conditions under which closed implies continuous.
Theorem 23.1.17 Let X and Y be Banach spaces and suppose L : X → Y is closed and linear. Then L is continuous.
Proof: Let G be the graph of L.G = {(x,Lx) : x ∈ X}. By Lemma 23.1.15 it follows that G is a Banach space. Define P : G → X by P(x,Lx) = x.P maps the Banach space G onto the Banach space X and is continuous and linear. By the open mapping theorem, P maps open sets onto open sets. Since P is also one to one, this says that P^{−1} is continuous. Thus

By Theorem 23.1.4 on Page 1590, this shows L is continuous. ■
The following corollary is quite useful. It shows how to obtain a new norm on the domain of a closed operator such that the domain with this new norm becomes a Banach space.
Corollary 23.1.18 Let L : D ⊆ X → Y where X,Y are a Banach spaces, and L is a closed operator. Then define a new norm on D by

Then D with this new norm is a Banach space.
Proof: If