# 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.

Contents 1 Theorem 2 Corollaries 3 Example: pointwise convergence of Fourier series 4 Generalizations 4.1 Barrelled spaces 4.2 Uniform boundedness in topological vector spaces 4.3 Generalizations involving nonmeager subsets 4.3.1 Complete metrizable domain 5 See also 6 Notes 7 Citations 8 Bibliography Theorem Uniform Boundedness Principle — Let {displaystyle X} be a Banach space, {displaystyle Y} a normed vector space and {displaystyle B(X,Y)} the space of all continuous linear operators from {displaystyle X} into {displaystyle Y} . Suppose that {displaystyle F} is a collection of continuous linear operators from {displaystyle X} to {displaystyle Y.} If {displaystyle sup _{Tin F}|T(x)|_{Y}0} such that {displaystyle {overline {B_{varepsilon }(x_{0})}}~:=~left{xin X,:,|x-x_{0}|leq varepsilon right}~subseteq ~X_{m}.} Let {displaystyle uin X} with {displaystyle |u|leq 1} and {displaystyle Tin F.} Then: {displaystyle {begin{aligned}|T(u)|_{Y}&=varepsilon ^{-1}left|Tleft(x_{0}+varepsilon uright)-Tleft(x_{0}right)right|_{Y}&[{text{ by linearity of }}T]\&leq varepsilon ^{-1}left(left|T(x_{0}+varepsilon u)right|_{Y}+left|T(x_{0})right|_{Y}right)\&leq varepsilon ^{-1}(m+m).&[{text{ since }} x_{0}+varepsilon u, x_{0}in X_{m}]\end{aligned}}} Taking the supremum over {displaystyle u} in the unit ball of {displaystyle X} and over {displaystyle Tin F} it follows that {displaystyle sup _{Tin F}|T|_{B(X,Y)}~leq ~2varepsilon ^{-1}m~<~infty .} There are also simple proofs not using the Baire theorem (Sokal 2011). Corollaries Corollary — If a sequence of bounded operators {displaystyle left(T_{n}right)} converges pointwise, that is, the limit of {displaystyle left(T_{n}(x)right)} exists for all {displaystyle xin X,} then these pointwise limits define a bounded linear operator {displaystyle T.} The above corollary does not claim that {displaystyle T_{n}} converges to {displaystyle T} in operator norm, that is, uniformly on bounded sets. However, since {displaystyle left{T_{n}right}} is bounded in operator norm, and the limit operator {displaystyle T} is continuous, a standard " {displaystyle 3varepsilon } " estimate shows that {displaystyle T_{n}} converges to {displaystyle T} uniformly on compact sets. Proof Essentially the same as that of the proof that a pointwise convergent sequence of uniformly continuous functions on a compact set converges to a continuous function. By uniform boundedness principle, let {displaystyle M=max{sup _{n}|T_{n}|,T}} be a uniform upper bound on the operator norms. Fix any compact {displaystyle Ksubset X} . Then for any {displaystyle epsilon >0} , finitely cover (use compactness) {displaystyle K} by a finite set of open balls {displaystyle {B(x_{i},r)}_{i=1,...,N}} of radius {displaystyle r={frac {epsilon }{M}}} Since {displaystyle T_{n}to T} pointwise on each of {displaystyle x_{1},...,x_{N}} , for all large {displaystyle n} , {displaystyle |T_{n}(x_{i})-T(x_{i})|leq epsilon } for all {displaystyle i=1,...,N} .

Then by triangle inequality, we find for all large {displaystyle n} , {displaystyle forall xin K,|T_{n}(x_{i})-T(x_{i})|leq 3epsilon } .

Corollary — Any weakly bounded subset {displaystyle Ssubseteq Y} in a normed space {displaystyle Y} is bounded.

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

Let {displaystyle L(X,Y)} denote the continuous operators from {displaystyle X} to {displaystyle Y,} endowed with the operator norm. If the collection {displaystyle F} is unbounded in {displaystyle L(X,Y),} then the uniform boundedness principle implies: {displaystyle R=left{xin X : sup nolimits _{Tin F}|Tx|_{Y}=infty right}neq varnothing .} In fact, {displaystyle R} is dense in {displaystyle X.} The complement of {displaystyle R} in {displaystyle X} is the countable union of closed sets {textstyle bigcup X_{n}.} By the argument used in proving the theorem, each {displaystyle X_{n}} is nowhere dense, i.e. the subset {textstyle bigcup X_{n}} is of first category. Therefore {displaystyle R} is the complement of a subset of first category in a Baire space. By definition of a Baire space, such sets (called comeagre or residual sets) are dense. Such reasoning leads to the principle of condensation of singularities, which can be formulated as follows: Theorem — Let {displaystyle X} be a Banach space, {displaystyle left(Y_{n}right)} a sequence of normed vector spaces, and for every {displaystyle n,} let {displaystyle F_{n}} an unbounded family in {displaystyle Lleft(X,Y_{n}right).} Then the set {displaystyle R:=left{xin X : {text{ for all }}nin mathbb {N} ,sup _{Tin F_{n}}|Tx|_{Y_{n}}=infty right}} is a residual set, and thus dense in {displaystyle X.} Proof The complement of {displaystyle R} is the countable union {displaystyle bigcup _{n,m}left{xin X : sup _{Tin F_{n}}|Tx|_{Y_{n}}leq mright}} of sets of first category. Therefore, its residual set {displaystyle R} is dense.

Example: pointwise convergence of Fourier series Let {displaystyle mathbb {T} } be the circle, and let {displaystyle C(mathbb {T} )} be the Banach space of continuous functions on {displaystyle mathbb {T} ,} with the uniform norm. Using the uniform boundedness principle, one can show that there exists an element in {displaystyle C(mathbb {T} )} for which the Fourier series does not converge pointwise.

For {displaystyle fin C(mathbb {T} ),} its Fourier series is defined by {displaystyle sum _{kin mathbb {Z} }{hat {f}}(k)e^{ikx}=sum _{kin mathbb {Z} }{frac {1}{2pi }}left(int _{0}^{2pi }f(t)e^{-ikt}dtright)e^{ikx},} and the N-th symmetric partial sum is {displaystyle S_{N}(f)(x)=sum _{k=-N}^{N}{hat {f}}(k)e^{ikx}={frac {1}{2pi }}int _{0}^{2pi }f(t)D_{N}(x-t),dt,} where {displaystyle D_{N}} is the {displaystyle N} -th Dirichlet kernel. Fix {displaystyle xin mathbb {T} } and consider the convergence of {displaystyle left{S_{N}(f)(x)right}.} The functional {displaystyle varphi _{N,x}:C(mathbb {T} )to mathbb {C} } defined by {displaystyle varphi _{N,x}(f)=S_{N}(f)(x),qquad fin C(mathbb {T} ),} is bounded. The norm of {displaystyle varphi _{N,x},} in the dual of {displaystyle C(mathbb {T} ),} is the norm of the signed measure {displaystyle (2(2pi )^{-1}D_{N}(x-t)dt,} namely {displaystyle left|varphi _{N,x}right|={frac {1}{2pi }}int _{0}^{2pi }left|D_{N}(x-t)right|,dt={frac {1}{2pi }}int _{0}^{2pi }left|D_{N}(s)right|,ds=left|D_{N}right|_{L^{1}(mathbb {T} )}.} It can be verified that {displaystyle {frac {1}{2pi }}int _{0}^{2pi }|D_{N}(t)|,dtgeq {frac {1}{2pi }}int _{0}^{2pi }{frac {left|sin left((N+{tfrac {1}{2}})tright)right|}{t/2}},dtto infty .} So the collection {displaystyle left(varphi _{N,x}right)} is unbounded in {displaystyle C(mathbb {T} )^{ast },} the dual of {displaystyle C(mathbb {T} ).} Therefore, by the uniform boundedness principle, for any {displaystyle xin mathbb {T} ,} the set of continuous functions whose Fourier series diverges at {displaystyle x} is dense in {displaystyle C(mathbb {T} ).} More can be concluded by applying the principle of condensation of singularities. Let {displaystyle left(x_{m}right)} be a dense sequence in {displaystyle mathbb {T} .} Define {displaystyle varphi _{N,x_{m}}} 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 {displaystyle x_{m}} is dense in {displaystyle C(mathbb {T} )} (however, the Fourier series of a continuous function {displaystyle f} converges to {displaystyle f(x)} for almost every {displaystyle xin mathbb {T} ,} by Carleson's theorem).

Generalizations In a topological vector space (TVS) {displaystyle X,} "bounded subset" refers specifically to the notion of a von Neumann bounded subset. If {displaystyle X} happens to also be a normed or seminormed space, say with (semi)norm {displaystyle |cdot |,} then a subset {displaystyle B} is (von Neumann) bounded if and only if it is norm bounded, which by definition means {textstyle sup _{bin B}|b|0} such that if {displaystyle tgeq r} then {displaystyle Csubseteq tU.} So for every {displaystyle hin H} and every {displaystyle tgeq r,} {displaystyle h(C)subseteq h(tU)=th(U)subseteq tV,} which implies that {textstyle bigcup _{hin H}h(C)subseteq tV.} Thus {textstyle bigcup _{hin H}h(C)} is bounded in {displaystyle Y.} Q.E.D. Citations ^ Shtern 2001. ^ Jump up to: a b c d Rudin 1991, pp. 42−47. Bibliography 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). Vol. 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]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. 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. Vol. 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. Vol. 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. Vol. 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. Vol. 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. show vte Functional analysis (topics – glossary) show vte Topological vector spaces (TVSs) Categories: Functional analysisMathematical principlesTheorems in functional analysis

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