Existing in Mathlib · Functional analysis
Closed Graph Theorem
A linear map between complete normed spaces cannot hide a discontinuity if its input-output graph already contains every one of its limit points: a closed graph forces the map to be continuous.
- Banach spaces
- closed graphs
- linear maps
- continuity
- complete subspaces
- continuous inverse theorem
Exact theorem
Exact Mathlib statement
theorem LinearMap.continuous_of_isClosed_graph {𝕜 E F : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (g : E →ₗ[𝕜] F) (hg : IsClosed (g.graph : Set (E × F))) : Continuous gThe theorem at a glance
The closed graph theorem at a glance
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
The closed graph theorem at a glance
Detailed visual description
The poster foregrounds the completeness assumptions, closed-graph hypothesis, and continuity conclusion. Convergent point trails terminate on one closed graph ribbon inside a product-space frame. Three lower stages treat the graph as complete, project it back to the source space, and recover the map from the continuous second coordinate. The footer keeps the forward-only, linear-map boundary explicit.
Statement structure
Statement and scope
Read the exact Mathlib declaration
This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.
A closed graph bridges source and target limits
Detailed visual description
Three vertically aligned convergence trails end at larger points in the pale-green source field. Each corresponds to one separated continuous cobalt ribbon across the enclosed central field, with every dotted trail terminating on its ribbon rather than at a gap. Two-way gold paths connect source and central endpoints, while one-way gold paths continue from the central endpoints to three gold convergence trails in the pale target field. The composition expresses only the forward closed-graph route and contains no inverse, compactness, or quantitative-bound claim.
Why it matters
A mathematical landmark
The closed graph theorem is one of functional analysis's foundational Banach-space principles. It replaces a direct continuity proof with a geometric condition on the input-output graph, and Mathlib's proof makes the complete graph space and continuous coordinate projections explicit.
ProofAtlas record
What has been checked
Mathlib is the source of the theorem; the local Lean replay and page review are separate.
Claim boundary
No new theorem is claimed
This page indexes Mathlib's closed graph declaration for an algebraic linear map between complete normed spaces over one nontrivially normed field. Closedness of graph(g) in E × F implies that g is continuous. The selected declaration is the forward implication only; it does not state an equivalence, a nonlinear theorem, compactness, bounded-inverse data, or an explicit operator-norm estimate.
- The selected declaration proves that a closed graph forces continuity; it does not package the converse into an if-and-only-if statement.
- The input g is an algebraic linear map between complete normed spaces, not an arbitrary nonlinear function and not a map assumed continuous in advance.
- The conclusion is function continuity. It does not provide an explicit operator norm, a quantitative continuity modulus, compactness, or bounded-inverse data.
- ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.
Source and local evidence
Where the theorem comes from
- Existing declaration
LinearMap.continuous_of_isClosed_graphin mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Local Lean evidence
artifact.library.mathlib.closed-graph-theorem.v001