# Uniform boundedness principle In Mathematik, 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.

Inhalt 1 Satz 2 Folgerungen 3 Beispiel: pointwise convergence of Fourier series 4 Verallgemeinerungen 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 Siehe auch 6 Anmerkungen 7 Zitate 8 Bibliography Theorem Uniform Boundedness Principle — Let {Anzeigestil X} be a Banach space, {Anzeigestil Y} a normed vector space and {Anzeigestil B(X,Y)} the space of all continuous linear operators from {Anzeigestil X} hinein {Anzeigestil Y} . Nehme an, dass {Anzeigestil F} is a collection of continuous linear operators from {Anzeigestil X} zu {Anzeigestil Y.} Wenn {displaystyle sup _{Tin F}|T(x)|_{Y}0} so dass {Anzeigestil {überstreichen {B_{varepsilon }(x_{0})}}~:=~left{xin X,:,|x-x_{0}|leq varepsilon right}~subseteq ~X_{m}.} Lassen {displaystyle uin X} mit {Anzeigestil |u|leq 1} und {displaystyle Tin F.} Dann: {Anzeigestil {Start{ausgerichtet}|T(u)|_{Y}&=varepsilon ^{-1}links|Tleft(x_{0}+varepsilon uright)-Tleft(x_{0}Rechts)Rechts|_{Y}&[{Text{ by linearity of }}T]\&leq varepsilon ^{-1}links(links|T(x_{0}+varepsilon u)Rechts|_{Y}+links|T(x_{0})Rechts|_{Y}Rechts)\&leq varepsilon ^{-1}(m+m).&[{Text{ seit }} x_{0}+varepsilon u, x_{0}in X_{m}]\Ende{ausgerichtet}}} Taking the supremum over {Anzeigestil u} in the unit ball of {Anzeigestil X} and over {displaystyle Tin F} es folgt dem {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) {Anzeigestil K} by a finite set of open balls {Anzeigestil {B(x_{ich},r)}_{i=1,...,N}} of radius {displaystyle r={frac {Epsilon }{M}}} Seit {Anzeigestil T_{n}to T} pointwise on each of {Anzeigestil x_{1},...,x_{N}} , for all large {Anzeigestil n} , {Anzeigestil |T_{n}(x_{ich})-T(x_{ich})|leq epsilon } für alle {displaystyle i=1,...,N} .

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

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

In der Tat, die Elemente von {Anzeigestil S} define a pointwise bounded family of continuous linear forms on the Banach space {Anzeigestil X:=Y',} which is the continuous dual space of {Anzeigestil Y.} By the uniform boundedness principle, the norms of elements of {Anzeigestil S,} as functionals on {Anzeigestil X,} das ist, 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 {Anzeigestil Y,} by a consequence of the Hahn–Banach theorem.

Lassen {Anzeigestil L(X,Y)} denote the continuous operators from {Anzeigestil X} zu {Anzeigestil Y,} endowed with the operator norm. If the collection {Anzeigestil F} is unbounded in {Anzeigestil 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 der Tat, {Anzeigestil R} is dense in {displaystyle X.} The complement of {Anzeigestil R} in {Anzeigestil X} is the countable union of closed sets {textstyle bigcup X_{n}.} By the argument used in proving the theorem, each {Anzeigestil X_{n}} is nowhere dense, d.h. die Teilmenge {textstyle bigcup X_{n}} is of first category. Deswegen {Anzeigestil 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 {Anzeigestil X} be a Banach space, {Anzeigestil links(Y_{n}Rechts)} a sequence of normed vector spaces, and for every {Anzeigestil n,} Lassen {Anzeigestil F_{n}} an unbounded family in {displaystyle Lleft(X,Y_{n}Rechts).} Then the set {Anzeigestil R:=links{xin X : {Text{ für alle }}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 {Anzeigestil R} is the countable union {displaystyle bigcup _{n,m}links{xin X : sup _{Tin F_{n}}|Tx|_{Y_{n}}leq mright}} of sets of first category. Deswegen, its residual set {Anzeigestil R} is dense.

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

Zum {displaystyle fin C(mathbb {T} ),} its Fourier series is defined by {Anzeigestil Summe _{kin mathbb {Z} }{Hut {f}}(k)e^{ikx}= Summe _{kin mathbb {Z} }{frac {1}{2Pi }}links(int _{0}^{2Pi }f(t)e^{-ikt}Richtig)e^{ikx},} and the N-th symmetric partial sum is {Anzeigestil S_{N}(f)(x)= Summe _{k=-N}^{N}{Hut {f}}(k)e^{ikx}={frac {1}{2Pi }}int _{0}^{2Pi }f(t)D_{N}(x-t),dt,} wo {displaystyle D_{N}} ist der {Anzeigestil N} -th Dirichlet kernel. Fix {displaystyle xin mathbb {T} } and consider the convergence of {Anzeigestil links{S_{N}(f)(x)Rechts}.} The functional {Anzeigestil Varphi _{N,x}:C(mathbb {T} )zu mathbb {C} } definiert von {Anzeigestil Varphi _{N,x}(f)=S_{N}(f)(x),qquad fin C(mathbb {T} ),} ist begrenzt. The norm of {Anzeigestil Varphi _{N,x},} in the dual of {Anzeigestil C(mathbb {T} ),} is the norm of the signed measure {Anzeigestil (2(2Pi )^{-1}D_{N}(x-t)dt,} nämlich {Anzeigestil links|varphi_{N,x}Rechts|={frac {1}{2Pi }}int _{0}^{2Pi }links|D_{N}(x-t)Rechts|,dt={frac {1}{2Pi }}int _{0}^{2Pi }links|D_{N}(s)Rechts|,ds=left|D_{N}Rechts|_{L^{1}(mathbb {T} )}.} It can be verified that {Anzeigestil {frac {1}{2Pi }}int _{0}^{2Pi }|D_{N}(t)|,dtgeq {frac {1}{2Pi }}int _{0}^{2Pi }{frac {links|sin left((N+{tfrac {1}{2}})tright)Rechts|}{t/2}},dtto infty .} So the collection {Anzeigestil links(varphi_{N,x}Rechts)} is unbounded in {Anzeigestil C(mathbb {T} )^{Ast },} the dual of {Anzeigestil C(mathbb {T} ).} Deswegen, by the uniform boundedness principle, für alle {displaystyle xin mathbb {T} ,} the set of continuous functions whose Fourier series diverges at {Anzeigestil x} is dense in {Anzeigestil C(mathbb {T} ).} More can be concluded by applying the principle of condensation of singularities. Lassen {Anzeigestil links(x_{m}Rechts)} be a dense sequence in {Anzeigestil mathbb {T} .} Definieren {Anzeigestil 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 {Anzeigestil x_{m}} is dense in {Anzeigestil C(mathbb {T} )} (jedoch, the Fourier series of a continuous function {Anzeigestil f} konvergiert zu {Anzeigestil f(x)} for almost every {displaystyle xin mathbb {T} ,} by Carleson's theorem).

Generalizations In a topological vector space (Fernseher) {Anzeigestil X,} "bounded subset" refers specifically to the notion of a von Neumann bounded subset. Wenn {Anzeigestil X} happens to also be a normed or seminormed space, say with (halb)Norm {Anzeigestil |cdot |,} then a subset {Anzeigestil B} ist (von Neumann) bounded if and only if it is norm bounded, which by definition means {textstyle sup _{bin B}|b|0} {displaystyle tgeq r} {displaystyle Csubseteq tU.} {displaystyle tgeq r,} {displaystyle h(C)subseteq h(tU)=th(U)subseteq tV,} {textstyle bigcup _{hin H}h(C)subseteq tV.} Citations Shtern 2001. Jump up to: b c d Rudin 1991, pp. 42−47. Bibliography Banach, Stefan; Steinhaus, Hugo (1927), "Sur le principe de la condensation singularités" (PDF), Fundamenta Mathematicae, 9: 50–61, doi:10.4064>

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