The Banach-Steinhaus theorem: Uniform Boundedness Principle #
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Herein we prove the Banach-Steinhaus theorem: any collection of bounded linear maps from a Banach space into a normed space which is pointwise bounded is uniformly bounded.
TODO #
For now, we only prove the standard version by appeal to the Baire category theorem.
Much more general versions exist (in particular, for maps from barrelled spaces to locally
convex spaces), but these are not yet in mathlib.
This is the standard Banach-Steinhaus theorem, or Uniform Boundedness Principle. If a family of continuous linear maps from a Banach space into a normed space is pointwise bounded, then the norms of these linear maps are uniformly bounded.
This version of Banach-Steinhaus is stated in terms of suprema of ↑‖⬝‖₊ : ℝ≥0∞
for convenience.
Given a sequence of continuous linear maps which converges pointwise and for which the domain is complete, the Banach-Steinhaus theorem is used to guarantee that the limit map is a continuous linear map as well.
Equations
- continuous_linear_map_of_tendsto g h = {to_linear_map := {to_fun := f, map_add' := _, map_smul' := _}, cont := _}