Existing in Mathlib · Functional analysis

Banach–Steinhaus Theorem

For each vector x, the family may need its own bound. If the source space is complete, those separate pointwise bounds force a single constant that bounds the operator norm of every map in the family.

uniform boundedness principle
continuous semilinear maps
operator norms
pointwise boundedness
equicontinuity
barrelled spaces

Exact theorem

Exact Mathlib statement

theorem banach_steinhaus {E F 𝕜 𝕜₂ : Type*} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {ι : Type*} [CompleteSpace E] {g : ι → E →SL[σ₁₂] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C'

The theorem at a glance

Banach–Steinhaus theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory functional-analysis poster states pointwise boundedness for a family of continuous semilinear maps, the resulting common operator-norm bound, and the checked route through equicontinuity and the barrelled-space theorem. — Pointwise boundedness on a complete source forces one uniform bound on every operator norm.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

For one fixed x, every shown output norm lies below the same ceiling C(x); an implication leads to all operator norms lying below one common ceiling C. — Vector-dependent pointwise bounds shared across the family yield one uniform operator-norm bound.

Why it matters

A mathematical landmark

Banach–Steinhaus is one of the foundational principles of functional analysis. It converts pointwise control of an entire operator family into uniform norm control, with completeness supplying the decisive global structure.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's Banach–Steinhaus theorem for an arbitrary family of continuous semilinear maps from a complete seminormed 𝕜-space E to a seminormed 𝕜₂-space F, over an isometric ring homomorphism between nontrivially normed fields. If the family is pointwise norm-bounded, then one real constant uniformly bounds every operator norm. The pointwise constant may depend on x. The declaration does not assert convergence, compactness, injectivity, surjectivity, a finite index type, or completeness of the codomain.

The pointwise hypothesis permits its bounding constant to depend on x; it does not assume one pointwise bound works for the whole domain.
The index type is arbitrary, and the declaration makes no finite-family, convergence, or compactness claim.
The selected scalar context allows continuous semilinear maps between spaces over different nontrivially normed fields linked by an isometric ring homomorphism.
Only the domain E is assumed complete; the codomain F is seminormed but need not carry a CompleteSpace instance.
Proof Atlas is indexing an existing Mathlib theorem, not claiming new mathematics, and the generated explanation and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: banach_steinhaus in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.banach-steinhaus-theorem.v001
Source: Open the pinned upstream reference