# Uniform boundedness principle

In mathematics, the **uniform boundedness principle** or **Banach–Steinhaus theorem** is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.

The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus but it was also proven independently by Hans Hahn.

## Theorem

**Theorem (Uniform Boundedness Principle).** Let *X* be a Banach space and *Y* be a normed vector space. Suppose that *F* is a collection of continuous linear operators from *X* to *Y*. If for all *x* in *X* one has

then

## Proof

The completeness of *X* enables the following short proof, using the Baire category theorem.

**Proof.** Suppose that for every *x* in the Banach space *X*, one has:

For every integer , let

The set is a closed set and by the assumption,

By the Baire category theorem for the non-empty complete metric space *X*, there exists *m* such that
has non-empty interior, *i.e.*, there exist and ε > 0 such that

Let *u* ∈ *X* with ǁ*u*ǁ ≤ 1 and *T* ∈ *F*. One has that:

Taking the supremum over *u* in the unit ball of *X*, it follows that

There are also simple proofs not using the Baire theorem (Sokal 2011).

## Corollaries

**Corollary.** If a sequence of bounded operators (*T _{n}*) converges pointwise, that is, the limit of {

*T*(

_{n}*x*)} exists for all

*x*in

*X*, then these pointwise limits define a bounded operator

*T*.

Note it is not claimed above that *T _{n}* converges to

*T*in operator norm, i.e. uniformly on bounded sets. (However, since {

*T*} is bounded in operator norm, and the limit operator

_{n}*T*is continuous, a standard "3-ε" estimate shows that

*T*converges to

_{n}*T*uniformly on

*compact*sets.)

**Corollary.** Any weakly bounded subset S in a normed space Y is bounded.

Indeed, the elements of *S* define a pointwise bounded family of continuous linear forms on the Banach space *X* = *Y**, continuous dual of *Y*. By the uniform boundedness principle, the norms of elements of *S*, as functionals on *X*, that is, norms in the second dual *Y***, are bounded. But for every *s* in *S*, the norm in the second dual coincides with the norm in *Y*, by a consequence of the Hahn–Banach theorem.

Let *L*(*X*, *Y*) denote the continuous operators from *X* to *Y*, with the operator norm. If the collection *F* is unbounded in *L*(*X*, *Y*), then by the uniform boundedness principle, we have:

In fact, *R* is dense in *X*. The complement of *R* in *X* is the countable union of closed sets ∪*X _{n}*. By the argument used in proving the theorem, each

*X*is nowhere dense, i.e. the subset ∪

_{n}*X*is

_{n}*of first category*. Therefore

*R*is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called

*residual sets*) are dense. Such reasoning leads to the

**principle of condensation of singularities**, which can be formulated as follows:

**Theorem.** Let *X* be a Banach space, {*Y _{n}*} a sequence of normed vector spaces, and

*F*a unbounded family in

_{n}*L*(

*X*,

*Y*). Then the set

_{n}is of second category, and thus dense in *X*.

**Proof.** The complement of *R* is the countable union

of sets of first category. Therefore its residual set *R* is dense.

## Example: pointwise convergence of Fourier series

Let be the circle, and let be the Banach space of continuous functions on , with the uniform norm. Using the uniform boundedness principle, one can show that there exists an element in for which the Fourier series does not converge pointwise.

For , its Fourier series is defined by

and the *N*-th symmetric partial sum is

where *D _{N}* is the

*N*-th Dirichlet kernel. Fix and consider the convergence of {

*S*(

_{N}*f*)(

*x*)}. The functional φ

_{N,x}: defined by

is bounded. The norm of φ_{N,x}, in the dual of , is the norm of the signed measure (2π)^{−1}*D*_{N}(*x*−*t*) d*t*, namely

One can verify that

So the collection {φ_{N,x}} is unbounded in , the dual of . Therefore by the uniform boundedness principle, for any , the set of continuous functions whose Fourier series diverges at *x* is dense in .

More can be concluded by applying the principle of condensation of singularities. Let {*x _{m}*} be a dense sequence in . Define φ

_{N,xm}in the similar way as above. The principle of condensation of singularities then says that the set of continuous functions whose Fourier series diverges at each

*x*is dense in (however, the Fourier series of a continuous function

_{m}*f*converges to

*f*(

*x*) for almost every , by Carleson's theorem).

## Generalizations

The least restrictive setting for the uniform boundedness principle is a barrelled space where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1):

**Theorem.** Given a barrelled space *X* and a locally convex space *Y*, then any family of pointwise bounded continuous linear mappings from *X* to *Y* is equicontinuous (even uniformly equicontinuous).

Alternatively, the statement also holds whenever *X* is a Baire space and *Y* is a locally convex space (Shtern 2001).

Dieudonné (1970) proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces. Specifically,

**Theorem.** Let *X* be a Fréchet space, *Y* a normed space, and *H* a set of continuous linear mappings of *X* into *Y*. If for every *x* in *X*

then the family *H* is equicontinuous.

## See also

- Barrelled space, a topological vector space with minimum requirements for the Banach Steinhaus theorem to hold

## References

- Banach, Stefan; Steinhaus, Hugo (1927), "Sur le principe de la condensation de singularités" (PDF),
*Fundamenta Mathematicae*,**9**: 50–61. (French) - Bourbaki, Nicolas (1987),
*Topological vector spaces*, Elements of mathematics, Springer, ISBN 978-3-540-42338-6 - Dieudonné, Jean (1970),
*Treatise on analysis, Volume 2*, Academic Press. - Rudin, Walter (1966),
*Real and complex analysis*, McGraw-Hill. - Shtern, A.I. (2001), "b/b015200", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4. - Sokal, Alan (2011), "A really simple elementary proof of the uniform boundedness theorem",
*Amer. Math. Monthly*,**118**: 450–452, arXiv:1005.1585, doi:10.4169/amer.math.monthly.118.05.450.