Closed graph arguments are useful when boundedness is hard to estimate directly. One proves stability of limits in the graph, and completeness upgrades that closedness to a norm bound.
Closed graphs
A graph remembers both an input and the corresponding output. Closedness says that if both coordinates converge, the limiting pair still satisfies the equation \(y=Tx\).
Definition. For normed spaces \(X,Y\), the graph of a linear map \(T:D(T)\subset X\to Y\) is \(G(T)=\{(x,Tx):x\in D(T)\}\subset X\times Y\). The operator is closed if \(G(T)\) is closed in \(X\times Y\).
Proposition. If \(D(T)\) is equipped with the graph norm \(\|x\|_T=\|x\|_X+\|Tx\|_Y\), then \(T\) is closed iff \(D(T)\) is complete whenever \(X,Y\) are Banach and \(D(T)\) is viewed with this graph norm.
Proof. The map \(x\mapsto(x,Tx)\) is an isometry from \((D(T),\|\cdot\|_T)\) onto \(G(T)\) up to the equivalent product norm \(\|(x,y)\|=\|x\|+\|y\|\). A subspace of the Banach product \(X\times Y\) is complete iff it is closed.
Closed graph theorem
The theorem converts a qualitative stability property into a quantitative operator bound. Completeness is needed on both coordinates so that the graph is itself a Banach space.
Theorem. Let \(X,Y\) be Banach spaces. If \(T:X\to Y\) is linear and its graph is closed in \(X\times Y\), then \(T\in\mathcal L(X,Y)\).
Proof. Since \(G(T)\) is a closed subspace of the Banach space \(X\times Y\), it is Banach. The projection \(\pi_1:G(T)\to X\), \(\pi_1(x,Tx)=x\), is bounded and bijective. By the bounded inverse theorem, \(\pi_1^{-1}:X\to G(T)\) is bounded. The second projection \(\pi_2:G(T)\to Y\) is bounded, and \(T=\pi_2\pi_1^{-1}\). Hence \(T\) is bounded.
Sequential closedness
Because normed spaces are metric, closedness of the graph can be checked by sequences. This is often the most practical form in analysis.
Proposition. For operators between normed spaces, \(T\) is closed iff whenever \(x_n\in D(T)\), \(x_n\to x\) in \(X\), and \(Tx_n\to y\) in \(Y\), one has \(x\in D(T)\) and \(Tx=y\).
Proof. This is exactly the sequential characterization of closed subsets in metric spaces applied to \(G(T)\subset X\times Y\). Convergence \((x_n,Tx_n)\to(x,y)\) in the product norm is equivalent to \(x_n\to x\) and \(Tx_n\to y\).
Closable operators
Unbounded operators may not have closed graphs initially. Closability asks whether the graph has a meaningful closed extension rather than losing uniqueness at the limit.
Definition. A densely defined operator \(T:D(T)\subset X\to Y\) is closable if \(\overline{G(T)}\) is the graph of an operator. Its closure is denoted \(\overline T\).
Proposition. A densely defined operator \(T\) is closable iff \(x_n\to0\) and \(Tx_n\to y\) imply \(y=0\).
Proof. If \(\overline{G(T)}\) is a graph, it cannot contain both \((0,0)\) and \((0,y)\) with \(y\ne0\). Conversely, the stated condition gives uniqueness of the second coordinate over each first coordinate in \(\overline{G(T)}\): if \((x,y_1)\) and \((x,y_2)\) lie in the closure, subtract approximating graph sequences to get a sequence tending to \(0\) whose image tends to \(y_1-y_2\), hence \(y_1=y_2\).
Differentiation as a closed operator
Differentiation is the canonical example where boundedness fails but closedness survives. The derivative is controlled by convergence of both functions and derivatives.
Example. On \(C[0,1]\) with the sup norm, define \(D:C^1[0,1]\to C[0,1]\) by \(Df=f'\). If \(f_n\to f\) uniformly and \(f_n'\to g\) uniformly, then \(f(t)-f(0)=\lim_n\int_0^t f_n'(s)\,\mathrm{d}s=\int_0^t g(s)\,\mathrm{d}s\). Thus \(f\in C^1\) and \(f'=g\), so \(D\) is closed. It is not bounded, because \(\|\sin(nt)\|_\infty\le1\) while \(\|n\cos(nt)\|_\infty=n\).