theorem UniformBoundedness.uniform_boundedness {V : Type u_1} [NormedAddCommGroup V] [NormedSpace ℝ V] [CompleteSpace V] {W : Type u_2} [NormedAddCommGroup W] [NormedSpace ℝ W] {A : Type u_3} {T : A → V →L[ℝ] W} (h : ∀ (v : V), ∃ (C : ℝ), ∀ (α : A), ‖(T α) v‖ ≤ C) : ∃ (C : ℝ), ∀ (α : A), ‖T α‖ ≤ C

Uniform Boundedness Theorem (Banach–Steinhaus). Let $B$ be a Banach space and let $\{T_\alpha\}_{\alpha \in A}$ be a family in $\mathcal{B}(B, V)$ (bounded linear operators from $B$ into a normed space $V$). If $\{T_\alpha\}$ is pointwise bounded, i.e. for every $v \in B$ there exists $C \in \mathbb{R}$ with $\|T_\alpha v\| \le C$ for all $\alpha$, then the operator norms are uniformly bounded: there exists $C \in \mathbb{R}$ with $\|T_\alpha\| \le C$ for all $\alpha$.