---
tag: YG0042V8CJ
kind: statement
role: theorem
name: Uniform boundedness principle
msc: [46B99, 46A30]
uses: [06GZ956657, 03J5RAKB30, 03TM135G0N, 06EKW1AKXF]
about: [03TM135G0N, 06EKW1AKXF]
sources:
  - url: https://en.wikipedia.org/wiki/Uniform_boundedness_principle
same_as:
  wikidata: Q1426292
  wikipedia: Uniform boundedness principle
---

## Statement

Let $X$ be a [[Banach space|06GZ956657]] and let $Y$ be a [[normed vector space|03J5RAKB30]], both over $\R$ or $\C$.
Let $\mathcal{F}$ be a family of [[continuous linear maps|03TM135G0N]] from $X$ to $Y$.
If $\sup_{T \in \mathcal{F}} \norm{Tx} < \infty$ for every $x \in X$, then $\sup_{T \in \mathcal{F}} \norm{T} < \infty$, where $\norm{T}$ is the [[operator norm|06EKW1AKXF]].

## Remarks

For $k \ge 1$ let $F_k = \set{x \in X : \norm{Tx} \le k \text{ for all } T \in \mathcal{F}}$.
Each $F_k$ is closed, and the hypothesis says that the $F_k$ cover $X$.
Since $X$ is complete, the [[Baire category theorem|0A7KGAYHGV]] gives some $F_k$ that contains a ball $B(x_0, r)$.
Then for $\norm{x} < r$ and $T \in \mathcal{F}$, $\norm{Tx} \le \norm{T(x_0 + x)} + \norm{T x_0} \le 2k$, so $\norm{T} \le 2k/r$.

A main use is the Banach–Steinhaus theorem: if continuous linear maps $T_n \colon X \to Y$ converge at every point to $Tx$, then $T$ is linear and continuous, since $\norm{Tx} \le \sup_n \norm{T_n}\,\norm{x}$.
Completeness of $X$ is necessary.
Let $X$ be the space of finitely supported real sequences with the norm $\norm{x}_\infty = \sup_k \abs{x_k}$, and let $T_n(x) = n x_n$.
For each $x$ the values $T_n x$ are eventually $0$, so they are bounded, but $\norm{T_n} = n$.

## Explanations

### L1 human

Suppose a family of linear operators on a complete space is bounded at each single point.
The theorem says that the family is then bounded uniformly, by one constant times the norm.
Pointwise control turns into global control, for free.

### L2 human

It is one of three consequences of the Baire category theorem at the core of a first course on Banach spaces.
The other two are the [[bounded inverse theorem|MR5FJXR348]] and the [[closed graph theorem|GXM8BX8YM7]].
A classical application shows that there is a continuous periodic function whose Fourier series diverges at a point: the partial-sum functionals at that point have unbounded norms.
It also shows that weakly convergent sequences in a normed space are bounded, by applying it to the sequence viewed in the bidual.