Uniform boundedness principle

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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

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Uniform Boundedness Principle — Let   be a Banach space,   a normed vector space and   the space of all continuous linear operators from   into  . Suppose that   is a collection of continuous linear operators from   to   If, for every  ,   then  

The first inequality (that is,   for all  ) states that the functionals in   are pointwise bounded while the second states that they are uniformly bounded. The second supremum always equals   and if   is not the trivial vector space (or if the supremum is taken over   rather than  ) then closed unit ball can be replaced with the unit sphere  

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

Proof

Suppose   is a Banach space and 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).

Corollaries

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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.

Proof

Essentially the same as that of the proof that a pointwise convergent sequence of equicontinuous functions on a compact set converges to a continuous function.

By uniform boundedness principle, let   be a uniform upper bound on the operator norms.

Fix any compact  . Then for any  , finitely cover (use compactness)   by a finite set of open balls   of radius  

Since   pointwise on each of  , for all large  ,   for all  .

Then by triangle inequality, we find for all large  ,  .

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 comeagre or 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 for every   let   an unbounded family in   Then the set   is a residual set, and thus dense in  

Proof

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

Example: pointwise convergence of Fourier series

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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).

Generalizations

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In a topological vector space (TVS)   "bounded subset" refers specifically to the notion of a von Neumann bounded subset. If   happens to also be a normed or seminormed space, say with (semi)norm   then a subset   is (von Neumann) bounded if and only if it is norm bounded, which by definition means  

Barrelled spaces

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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]

Uniform boundedness in topological vector spaces

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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   In particular, if   is a family of maps from   to   and if   then the family   is uniformly bounded in   if and only if there exists some bounded subset   of   such that   which happens if and only if   is a bounded subset of  

Proposition[2] — Let   be a set of continuous linear operators between two topological vector spaces   and   and let   be any bounded subset of   Then the family of sets   is uniformly bounded in   if any of the following conditions are satisfied:

  1.   is equicontinuous.
  2.   is a convex compact Hausdorff subspace of   and for every   the orbit   is a bounded subset of  

Generalizations involving nonmeager subsets

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Although the notion of a nonmeager set is used in the following version of the uniform bounded principle, 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.

Every proper vector subspace of a TVS   has an empty interior in  [3] So in particular, every proper vector subspace that is closed is nowhere dense in   and thus of the first category (meager) in   (and the same is thus also true of all its subsets). Consequently, any vector subspace of a TVS   that is of the second category (nonmeager) in   must be a dense subset of   (since otherwise its closure in   would a closed proper vector subspace of   and thus of the first category).[3]

Proof[2]

Proof that   is equicontinuous:

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 not 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 not empty. Pick   Because the map   defined by   is a homeomorphism, the set   is a neighborhood of   in   which implies that the same is true of its superset   And so for every     This proves that   is equicontinuous. Q.E.D.


Proof that  :

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   Q.E.D.

Sequences of continuous linear maps

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The following theorem establishes conditions for the pointwise limit of a sequence of continuous linear maps to be itself continuous.

Theorem[4] — Suppose that   is a sequence of continuous linear maps between two topological vector spaces   and  

  1. If the set   of all   for which   is a Cauchy sequence in   is of the second category in   then  
  2. If the set   of all   at which the limit   exists in   is of the second category in   and if   is a complete metrizable topological vector space (such as a Fréchet space or an F-space), then   and   is a continuous linear map.

Theorem[3] — If   is a sequence of continuous linear maps from an F-space   into a Hausdorff topological vector space   such that for every   the limit   exists in   then   is a continuous linear map and the maps   are equicontinuous.

If in addition the domain is a Banach space and the codomain is a normed space then  

Complete metrizable domain

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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 or an F-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.

See also

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  • Barrelled space – Type of topological vector space
  • Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Notes

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Citations

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  1. ^ Shtern 2001.
  2. ^ a b c d Rudin 1991, pp. 42−47.
  3. ^ a b c Rudin 1991, p. 46.
  4. ^ Rudin 1991, pp. 45−46.

Bibliography

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