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.

- Theorem
- Corollaries
- Example: pointwise convergence of Fourier series
- Generalizations
- Barrelled spaces
- Uniform boundedness in general topological vector spaces
- Generalizations involving nonmeager subsets
- See also
- Notes
- Citations
- Bibliography

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

**Uniform Boundedness Principle** — Let be a Banach space and a normed vector space. Suppose that is a collection of continuous linear operators from to If

then

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

Proof |
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Let X be a Banach space. Suppose that for every For every integer let Each set is a closed set and by the assumption, By the Baire category theorem for the non-empty complete metric space there exists some such that has non-empty interior; that is, there exist and such that Let with and Then: Taking the supremum over in the unit ball of and over it follows that |

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

**Corollary** — If a sequence of bounded operators converges pointwise, that is, the limit of exists for all then these pointwise limits define a bounded linear operator

The above corollary does *not* claim that converges to in operator norm, that is, uniformly on bounded sets. However, since is bounded in operator norm, and the limit operator is continuous, a standard "" estimate shows that converges to uniformly on *compact* sets.

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

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

Let denote the continuous operators from to endowed with the operator norm. If the collection is unbounded in then the uniform boundedness principle implies:

In fact, is dense in The complement of in is the countable union of closed sets By the argument used in proving the theorem, each is nowhere dense, i.e. the subset is *of first category*. Therefore 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 be a Banach space, a sequence of normed vector spaces, and an unbounded family in Then the set

is a residual set, and thus dense in

Proof |
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The complement of is the countable union of sets of first category. Therefore, its residual set is dense. |

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 is the -th Dirichlet kernel. Fix and consider the convergence of The functional defined by

is bounded. The norm of in the dual of is the norm of the signed measure namely

It can be verified that

So the collection is unbounded in the dual of Therefore, by the uniform boundedness principle, for any the set of continuous functions whose Fourier series diverges at is dense in

More can be concluded by applying the principle of condensation of singularities. Let be a dense sequence in Define 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 is dense in (however, the Fourier series of a continuous function converges to for almost every by Carleson's theorem).

Attempts to find classes of locally convex topological vector spaces on which the uniform boundedness principle holds eventually led to barrelled spaces. That is, 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 and a locally convex space then any family of pointwise bounded continuous linear mappings from to is equicontinuous (and even uniformly equicontinuous).

Alternatively, the statement also holds whenever is a Baire space and is a locally convex space.^{ [1] }

A family of subsets of a topological vector space is said to be *uniformly bounded* in if there exists some bounded subset of such that

which happens if and only if

is a bounded subset of ; if is a normed space then this happens if and only if there exists some real such that

The below theorem's conclusion that the set is necessarily equal to all to can be deduced, with the help of the first part of the following theorem, from the equicontinuity of and the fact that every single subset of is also a bounded subset.

**Theorem ^{ [2] }** — Let be a set of continuous linear operators between two topological vector spaces and and let be any bounded subset of If is equicontinuous then the family of sets is uniformly bounded in

Also, if is a convex compact Hausdorff subspace and if for every the orbit is a bounded subset of then the family is uniformly bounded in (for this conclusion, was not assumed to be equicontinuous).

In the following version of the theorem, the domain is not assumed to be a Baire space.

**Theorem ^{ [2] }** — Let be a set of continuous linear operators between two topological vector spaces and (not necessarily Hausdorff or locally convex). For every denote the orbit of by

and let denote the set of all whose orbit is a bounded subset of If is of the second category (that is, nonmeager) in then and is equicontinuous.

Proof ^{ [2] } |
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Let be balanced neighborhoods of the origin in satisfying It must be shown that there exists a neighborhood of the origin in such that for every Let which is a closed subset of (because it is an intersection of closed subsets) that for every also satisfies and (as will be shown, the set is in fact a neighborhood of the origin in because the topological interior of in is non-empty). If then being bounded in implies that there exists some integer such that so if then Since was arbitrary, This proves that Because is of the second category in the same must be true of at least one of the sets for some The map defined by is a (surjective) homeomorphism, so the set is necessarily of the second category in Because is closed and of the second category in its topological interior in is non-empty. Pick The map defined by being a homeomorphism implies that is a neighborhood of in (which implies that the same is true of the superset ). And so for every This proves that is equicontinuous.
Because is equicontinuous, if is bounded in then is uniformly bounded in In particular, for any because is a bounded subset of is a uniformly bounded subset of Thus |

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

**Theorem ^{ [2] }** — Let be a set of continuous linear operators from a complete metrizable topological vector space (such as a Fréchet space) into a Hausdorff topological vector space If for every the orbit

is a bounded subset of then is equicontinuous.

So in particular, if is also a normed space and if

then is equicontinuous.

- Barrelled space – A topological vector space with near minimum requirements for the Banach–Steinhaus theorem to hold.
- Ursescu theorem – Theorem that simultaneously generalizes the closed graph, open mapping, and Banach–Steinhaus theorems

- ↑ Let be a neighborhood of the origin in Since is equicontinuous, there exists a neighborhood of the origin in such that for every Because is bounded in there exists some real such that if then So for every and every which implies that Thus is bounded in

- ↑ Shtern 2001.
- 1 2 3 4 Rudin 1991, pp. 42−47.

- Banach, Stefan; Steinhaus, Hugo (1927), "Sur le principe de la condensation de singularités" (PDF),
*Fundamenta Mathematicae*,**9**: 50–61, doi: 10.4064/fm-9-1-50-61 . (in French) - Banach, Stefan (1932).
*Théorie des Opérations Linéaires*[*Theory of Linear Operations*](PDF). Monografie Matematyczne (in French).**1**. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11. - Bourbaki, Nicolas (1987) [1981].
*Sur certains espaces vectoriels topologiques*[*Topological Vector Spaces: Chapters 1–5*].*Annales de l'Institut Fourier*. Éléments de mathématique.**2**. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. - Dieudonné, Jean (1970),
*Treatise on analysis, Volume 2*, Academic Press. - Husain, Taqdir; Khaleelulla, S. M. (1978).
*Barrelledness in Topological and Ordered Vector Spaces*. Lecture Notes in Mathematics.**692**. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. - Khaleelulla, S. M. (1982).
*Counterexamples in Topological Vector Spaces*. Lecture Notes in Mathematics.**936**. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Rudin, Walter (1966),
*Real and complex analysis*, McGraw-Hill. - Rudin, Walter (1991).
*Functional Analysis*. International Series in Pure and Applied Mathematics.**8**(Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Schechter, Eric (1996).
*Handbook of Analysis and Its Foundations*. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. - Shtern, A.I. (2001) [1994], "Uniform boundedness principle",
*Encyclopedia of Mathematics*, EMS Press . - Sokal, Alan (2011), "A really simple elementary proof of the uniform boundedness theorem",
*Amer. Math. Monthly*,**118**(5): 450–452, arXiv: 1005.1585 , doi:10.4169/amer.math.monthly.118.05.450, S2CID 41853641 . - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. - Wilansky, Albert (2013).
*Modern Methods in Topological Vector Spaces*. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is *not* reflexive but is nevertheless isometrically isomorphic to its bidual.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In functional analysis, a **bounded linear operator** is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces, then is bounded if and only if there exists some such that for all in

In mathematical analysis, a family of functions is **equicontinuous** if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus *sequences* of functions.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

The **Arzelà–Ascoli theorem** is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The **transpose** or **algebraic adjoint** of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors.

In topology and related branches of mathematics, **total-boundedness** is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size"

In functional analysis and related areas of mathematics, a **barrelled space** is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A **barrelled set** or a **barrel** in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In functional and convex analysis, and related disciplines of mathematics, the **polar set** is a special convex set associated to any subset of a vector space lying in the dual space The **bipolar** of a subset is the polar of but lies in .

In functional analysis and related areas of mathematics a **polar topology**, **topology of -convergence** or **topology of uniform convergence on the sets of** is a method to define locally convex topologies on the vector spaces of a pairing.

In mathematics, a **nuclear space** is a topological vector space that can be viewed as a generalization of finite dimensional Euclidean spaces that is different from Hilbert spaces. Nuclear spaces have many of the desirable properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.

In mathematics, a linear map is a mapping between two modules that preserves the operations of addition and scalar multiplication.

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In mathematics, the **injective tensor product** of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the *completed injective tensor products*. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y with*out* any need to extend definitions from real/complex-valued functions to Y-valued functions.

In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

In functional analysis, every C^{*}-algebra is isomorphic to a subalgebra of the C^{*}-algebra of bounded linear operators on some Hilbert space *H*. This article describes the spectral theory of closed normal subalgebras of

This is a glossary for the terminology in a mathematical field of functional analysis.

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