Closed Graph Theorem

Statement

Let $E$ and $F$ be Banach spaces and $T : E \to F$ a linear map. If the graph of $T$,

is a closed subset of $E \times F$, then $T$ is bounded (continuous).

Closedness of the graph means: whenever $x_n \to x$ in $E$ and $Tx_n \to y$ in $F$, it follows that $y = Tx$.

Visualization

Trace the sequence criterion for a $2 \times 2$ matrix operator $T : \mathbb{R}^2 \to \mathbb{R}^2$, $Tv = Av$:

Sequence: x_n → x in ℝ²,   T(x_n) → y in ℝ²

x_n = (1 + 1/n, 2 + 1/n)  →  x = (1, 2)

A = <a class="concept-link" data-concept="3, 0], [0, 1">3, 0], [0, 1</a>
T(x_n) = (3 + 3/n, 2 + 1/n)  →  (3, 2) = T(1, 2) = y   ✓

graph(T) closed: (x_n, T(x_n)) → (x, y) and y = Tx ✓

$n$	$x_n$	$Tx_n$	limit
1	$(2, 3)$	$(6, 3)$	—
2	$(1.5, 2.5)$	$(4.5, 2.5)$	—
5	$(1.2, 2.2)$	$(3.6, 2.2)$	—
$\infty$	$(1, 2)$	$(3, 2)$	$Tx$

The limit point $(1,2), (3,2)$ lies on the graph: $T(1,2) = (3,2)$. Graph is closed, so $T$ is continuous.

Contrast: $T(x) = 1/x$ on $(0,1)$ has a non-closed graph (the point $(0, \infty)$ is a limit not in the graph), and $T$ is not continuous at $0$.

Proof Sketch

1. Product Banach space. $E \times F$ with norm $\|(x,y)\| = \max(\|x\|, \|y\|)$ is a Banach space.
2. Graph is a subspace. $G = \mathrm{graph}(T)$ is a linear subspace of $E \times F$.
3. $G$ is complete. A closed subspace of a Banach space is itself a Banach space (complete).
4. Projection maps. The projection $\pi_1 : G \to E$, $\pi_1(x, Tx) = x$, is a bijective bounded linear map.
5. Open Mapping Theorem. $\pi_1$ is bijective between Banach spaces, so by the Open Mapping TheoremOpen Mapping Theorem (Banach)$$T : E \twoheadrightarrow F \text{ bounded, surjective} \Rightarrow T \text{ is an open map}$$A surjective bounded linear map between Banach spaces maps open sets to open setsRead more →, its inverse $\pi_1^{-1} : E \to G$ is bounded.
6. Continuity of $T$. Write $T = \pi_2 \circ \pi_1^{-1}$; both factors are bounded.

Connections

• Open Mapping Theorem (Banach)Open Mapping Theorem (Banach)$$T : E \twoheadrightarrow F \text{ bounded, surjective} \Rightarrow T \text{ is an open map}$$A surjective bounded linear map between Banach spaces maps open sets to open setsRead more → — the Closed Graph Theorem is deduced directly from it via the projection argument.
• Hahn–Banach TheoremHahn–Banach Theorem$$\exists\, g : E \to \mathbb{R},\; g|_p = f,\; \|g\| = \|f\|$$A bounded linear functional on a subspace of a normed space extends to the whole space with the same normRead more → — together with the Open Mapping Theorem and Uniform Boundedness, these four results form the four pillars of functional analysis.
• Bolzano–Weierstrass TheoremBolzano–Weierstrass Theorem$$\text{bounded } (x_n) \subseteq \mathbb{R}^n \implies \exists\, \text{convergent subsequence}$$Every bounded sequence in ℝⁿ has a convergent subsequenceRead more → — the sequential characterisation of closed sets used here is the infinite-dimensional cousin of Bolzano–Weierstrass convergence.
• Spectral TheoremSpectral Theorem$$A = Q \Lambda Q^T \;\text{(real symmetric case)}$$Every real symmetric matrix is orthogonally diagonalizableRead more → — the Closed Graph Theorem ensures that many unbounded operators (e.g., differential operators with closed graph) still admit spectral decompositions.

import Mathlib.Analysis.Normed.Operator.Banach

/-- **Closed Graph Theorem**: a linear map with closed graph between Banach spaces is continuous.
    Direct alias of `LinearMap.continuous_of_isClosed_graph` in Mathlib. -/
theorem closed_graph_theorem {E F : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
    [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
    (g : E →ₗ[ℝ] F) (hg : IsClosed (g.graph : Set (E × F))) :
    Continuous g :=
  g.continuous_of_isClosed_graph hg