Uniform Boundedness Principle

Statement

(Banach–Steinhaus Theorem.) Let $E$ be a Banach space, $F$ a normed space, and $\{T_\iota\}_{\iota \in I}$ a family of bounded linear operators $T_\iota : E \to F$. If

then the operator norms are uniformly bounded:

Pointwise boundedness (a condition on each vector $x$) automatically upgrades to a uniform bound on all operators.

Visualization

Counterexample anatomy — why completeness is essential. On $c_{00}$ (finitely supported sequences, incomplete), define $T_n(x) = n x_n$:

x = (x₁, x₂, x₃, ...) ∈ c₀₀

T_n(x) = n · x_n

Pointwise: for each fixed x, only finitely many x_n ≠ 0,
           so sup_n |T_n(x)| = sup_n n|x_n| < ∞.

But ‖T_n‖ = n → ∞.   (Not uniformly bounded!)

On a complete space this cannot happen. Numerical trace on $\ell^2$ with a convergent family:

$n$	$T_n$ (truncation to first $n$ coords)	$\|T_n x\|$ for $x = (1/k^2)$	$\|T_n\|$
1	$(x_1, 0, 0, \ldots)$	$1$	$1$
2	$(x_1, x_2, 0, \ldots)$	$\sqrt{1 + 1/16}$	$1$
5	projection to 5 coords	$\sqrt{\sum_{k=1}^5 k^{-4}}$	$1$
$\infty$	identity	$\pi^2/6$	$1$

Here $\sup_n \|T_n\| = 1 < \infty$, consistent with UBP. The principle would fire if any $\sup_n \|T_n x\|$ were infinite — on a Banach space that cannot happen pointwise.

Proof Sketch

1. Define sets. For each $M > 0$ let $A_M = \{x \in E : \sup_\iota \|T_\iota x\| \le M\}$.
2. Closed sets. Each $A_M$ is closed (intersection of closed sets $\{x : \|T_\iota x\| \le M\}$).
3. Baire's theorem. Since $E = \bigcup_{M \in \mathbb{N}} A_M$ and $E$ is complete, some $A_M$ has non-empty interior: $B_r(x_0) \subseteq A_M$.
4. Symmetry trick. For any $y \in B_r(0)$ and any $\iota$:

5. Scale. Any $x$ with $\|x\| \le 1$ can be rescaled into $B_r(0)$, giving $\|T_\iota x\| \le 2M/r$ for all $\iota$.

Connections

• Closed Graph TheoremClosed Graph Theorem$$\mathrm{graph}(T) \text{ closed} \Rightarrow T \text{ continuous}$$A linear map between Banach spaces with closed graph is automatically continuousRead more → — both theorems use Baire's category theorem on Banach spaces; UBP uses it for a family, Closed Graph for a single operator.
• Cauchy–Schwarz InequalityCauchy–Schwarz Inequality$$|\langle u, v \rangle| \leq \|u\| \, \|v\|$$The inner product of two vectors is bounded by the product of their normsRead more → — in Hilbert spaces, uniform boundedness combines with Cauchy–Schwarz to bound sesquilinear forms.
• Spectral Radius FormulaSpectral Radius Formula$$r(a) = \lim_{n\to\infty} \|a^n\|^{1/n}$$The spectral radius equals the limit of the nth root of the operator norm of the nth power (Gelfand's formula)Read more → — operator norms bounded by the spectral radius formula (Gelfand) give a quantitative form of UBP for Banach algebras.
• Monotone Convergence TheoremMonotone Convergence Theorem$$f \text{ monotone},\; \sup_n f(n) < \infty \implies f(n) \to \sup_n f(n)$$A bounded monotone sequence of reals always converges — the supremum is the limitRead more → — in measure theory, both MCT and UBP serve as "upgrade" theorems: pointwise properties become uniform ones.

import Mathlib.Analysis.Normed.Operator.BanachSteinhaus

/-- **Banach–Steinhaus / Uniform Boundedness Principle**:
    a pointwise-bounded family of continuous linear maps on a Banach space
    has uniformly bounded norms. Direct alias of `banach_steinhaus`. -/
theorem uniform_boundedness_principle {ι : Type*} {E F : Type*}
    [SeminormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
    [SeminormedAddCommGroup F] [NormedSpace ℝ F]
    (g : ι → E →L[ℝ] F)
    (h : ∀ x, BddAbove (Set.range fun i => ‖g i x‖)) :
    BddAbove (Set.range fun i => ‖g i‖) :=
  banach_steinhaus h