Baire's Theorem

The set of Baire , even Baire category set , set of Baire-Hausdorff or simply category set called, is a theorem from mathematics . It is formulated in different versions in the literature and essentially contains a topological statement. This statement is of considerable importance in various related sub-areas of mathematics such as descriptive set theory , measure theory and functional analysis . Both the Banach-Steinhaus theorem , theDerive the principle of uniform limitedness and the theorem on open mapping from Baire's theorem. The naming as "category set" is based on the fact that special quantities are used for the formulation of the set, which are referred to as sets of the first and sets of second category . There is no direct reference to category theory .

The first versions of the theorem are by William Fogg Osgood (1897 derivation for the special case of the real axis ) and René Louis Baire (1899 derivation for the special case of Euclidean space ). A more general version was shown by Felix Hausdorff in 1914. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Preliminary remark

A few terms are necessary to formulate Baire's theorem. They are briefly defined below; a detailed description with examples and comments can be found in the corresponding main article.

A crowd is nowhere called dense when the inside of its closure is empty.
A set is called a lean set if it is a countable union of nowhere dense sets. Lean sets are also synonymous with sets of the first (Baire) category.
A quantity is called a fat quantity if it is not lean. Fat quantities are also synonymous with quantities of the second (baire) category.
A quantity is called a lesser quantity when it is the complement of a lean quantity. More coarse sets are synonymous with residual sets.

statement

Equivalent statements with alternative formulations using the terms listed above can be found in brackets.

Be a full metric space . Then the following four (equivalent) statements hold and are called Baire's Theorem: ${\ displaystyle (X, d)}$

The intersection of countably many dense , open sets is dense again.
Every open, non-empty subset is bold (every open, non-empty set is of the second category)
The complement of a lean set is dense in (Every lean set in is dense in ) ${\ displaystyle X}$ ${\ displaystyle X}$ ${\ displaystyle X}$
Every union of countably many closed sets without internal points has no internal point.

The following, weaker statement is also sometimes referred to as Baire's theorem:

If there is a non-empty complete metric space and if there are closed sets that cover, then one exists such that it has a non-empty interior (a non-empty complete metric space is of the second category in itself) ${\ displaystyle X}$ ${\ displaystyle (A_ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle X}$ ${\ displaystyle k_ {0}}$ ${\ displaystyle A_ {k_ {0}}}$

More generally, the following statement is called Baire's Theorem:

Every fully metrizable room and every locally compact Hausdorff room is a Baire room

Applications

Baire's theorem enables elegant proofs of central theorems of classical functional analysis:

The existence of nowhere differentiable functions

To exist continuous functions which are differentiable at any point. To see this, you bet for ${\ displaystyle [0,1]}$ ${\ displaystyle n \ in \ mathbb {N}}$

{\ displaystyle O_ {n}: = \ left \ {f \ in C [0,1] \ \ left | \ \ forall t \ in [0,1] \ colon \ sup _ {0 <| h | <{ \ frac {1} {n}}} \ left | {\ frac {f (t + h) -f (t)} {h}} \ right |> n \ right. \ right \}.}

If one provides the vector space with the supremum norm , one can show that it is open and dense . Because of Baire's theorem, we know that space is close in . The functions in are continuous and cannot be differentiated at any point. ${\ displaystyle C ([0,1])}$ ${\ displaystyle O_ {n}}$ ${\ displaystyle C ([0,1])}$ ${\ displaystyle \ textstyle D: = \ bigcap _ {n = 1} ^ {\ infty} O_ {n}}$ ${\ displaystyle C ([0,1])}$ ${\ displaystyle D}$

Basis of a Banach space

Another application of Baire's theorem shows that every basis of an infinite-dimensional Banach space is uncountable.

Proof by the counter-assumption that there is a countable basis of the Banach space . Be . Then: ${\ displaystyle \ left \ {b_ {n} \ right \} _ {n \ in \ mathbb {N}}}$ ${\ displaystyle \ left.V \ right.}$ ${\ displaystyle V_ {n}: = \ mathrm {span} \ left \ {b_ {1}, b_ {2}, \ dots, b_ {n} \ right \}}$

as finite-dimensional vector spaces are closed, ${\ displaystyle \ left.V_ {n} \ right.}$
their union gives the whole room: . ${\ displaystyle \ cup _ {n \ in \ mathbb {N}} \, V_ {n} = V}$

According to Baire's theorem, one of the must contain a ball. A sub-vector space that contains a sphere is always the whole space. This would result in a finite dimensional space, which leads to a contradiction. ${\ displaystyle \ left.V_ {n} \ right.}$ ${\ displaystyle \ left.V \ right.}$

Countable locally compact topological groups

With Baire's theorem it can be shown that at most countable locally compact, Hausdorff topological groups are discrete : They are the union of at most countable many one-element sets. These are closed, so according to Baire's theorem, at least one of them must be open. This means that there is an isolated point in the group , but this also isolates all points, since topological groups are homogeneous and the topology is discrete.

Comparable concept formation in measure theory

In measure theory it is shown that the space provided with the Hausdorff or Lebesgue measure cannot be written as a countable union of zero sets. If the term zero set is replaced by a lean set , then in this special case one receives the statement of the Baie set of categories . The Baie categories can thus be seen as a topological analogue to zero sets or measure spaces in measure theory. Indeed, they have a lot in common. These are described in detail in Oxtoby (1980). Note, however, that there are meager sets that are not zero sets and vice versa. ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Individual evidence

↑ ^a ^b Dirk Werner : functional analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , pp. 139 , doi : 10.1007 / 978-3-642-21017-4 .
↑ ^a ^b Winfried Kaballo : Basic Course Functional Analysis . 1st edition. Spektrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2149-4 , pp. 144-145 .
^ Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , p. 174-176 , doi : 10.1007 / 978-3-642-56860-2 .
^ Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , p. 229-333 , doi : 10.1007 / 978-3-642-22261-0 .
^ Bourbaki: Elements of Mathematics. General Topology. Part 2. 1966, pp. 193-194, 272-273.
↑ ^a ^b Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , p. 173-174 , doi : 10.1007 / 978-3-642-56860-2 .
^ Manfred Dobrowolski: Applied functional analysis . Functional analysis, Sobolev spaces and elliptic differential equations. 2nd, corrected and revised edition. Springer-Verlag, Berlin / Heidelberg 2010, ISBN 978-3-642-15268-9 , pp. 43 , doi : 10.1007 / 978-3-642-15269-6 .
↑ Winfried Kaballo : Basic Course Functional Analysis . 1st edition. Spektrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2149-4 , pp. 145 .
↑ Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , pp. 140 , doi : 10.1007 / 978-3-642-21017-4 .
^ Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , p. 229 , doi : 10.1007 / 978-3-642-22261-0 .
↑ Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , pp. 141 , doi : 10.1007 / 978-3-642-21017-4 .
↑ Reinhold Meise, Dietmar Vogt: Introduction to Functional Analysis. Vieweg, Braunschweig et al. 1992, ISBN 3-528-07262-8 , Chapter 1, § 8: Conclusions from Baire's theorem.

literature

René Baire : Sur les fonctions de variables réelles . In: Annali di Matematica Pura ed Applicata . tape 3 , no. 1 , 1899, p. 1–123 , doi : 10.1007 / BF02419243 .
Nicolas Bourbaki : Elements of Mathematics. General Topology. Part 2 (= ADIWES International Series in Mathematics ). Addison-Wesley Publishing Company, Reading MA et al. 1966.
William F. Osgood : Non-Uniform Convergence and the Integration of Series Term by Term . In: American Journal of Mathematics . tape 19 , no. 2 , 1897, p. 155-190 , doi : 10.2307 / 2369589 .
John C. Oxtoby: Measure and Category. A Survey of the Analogies between topological and measure spaces (= Graduate Texts in Mathematics. 2). 2nd edition. Springer, New York NY et al. 1980, ISBN 3-540-90508-1
Horst Schubert : Topology. An introduction . 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .
Dirk Werner : Functional Analysis. 5th enlarged edition. Springer, Berlin et al. 2005, ISBN 3-540-21381-3 .

[Werner139-1] Dirk Werner : functional analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , pp. 139 , doi : 10.1007 / 978-3-642-21017-4 .

[Kaballo1144-2] Winfried Kaballo : Basic Course Functional Analysis . 1st edition. Spektrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2149-4 , pp. 144-145 .

[vonQuerenburg174176-3] Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , p. 174-176 , doi : 10.1007 / 978-3-642-56860-2 .

[Alt229333-4] Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , p. 229-333 , doi : 10.1007 / 978-3-642-22261-0 .

[Bourbaki-5] Bourbaki: Elements of Mathematics. General Topology. Part 2. 1966, pp. 193-194, 272-273.

[Boto173-6] Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , p. 173-174 , doi : 10.1007 / 978-3-642-56860-2 .

[Dobr43-7] Manfred Dobrowolski: Applied functional analysis . Functional analysis, Sobolev spaces and elliptic differential equations. 2nd, corrected and revised edition. Springer-Verlag, Berlin / Heidelberg 2010, ISBN 978-3-642-15268-9 , pp. 43 , doi : 10.1007 / 978-3-642-15269-6 .

[Kaballo145-8] Winfried Kaballo : Basic Course Functional Analysis . 1st edition. Spektrum Akademischer Verlag, Heidelberg 2011, ISBN 978-3-8274-2149-4 , pp. 145 .

[Werner140-9] Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , pp. 140 , doi : 10.1007 / 978-3-642-21017-4 .

[Alt229-10] Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , p. 229 , doi : 10.1007 / 978-3-642-22261-0 .

[Werner141-11] Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , pp. 141 , doi : 10.1007 / 978-3-642-21017-4 .

[12] Reinhold Meise, Dietmar Vogt: Introduction to Functional Analysis. Vieweg, Braunschweig et al. 1992, ISBN 3-528-07262-8 , Chapter 1, § 8: Conclusions from Baire's theorem.