В функциональном анализе и смежных отраслей математики , то теорема Банаха-Алаоглу (также известный как теорема Алаоглу ) утверждает , что замкнутый единичный шар в сопряженном пространстве в виде нормированного векторного пространства является компактным в слабой * топологии . [1] Обычное доказательство идентифицирует единичный шар со слабой * топологией как замкнутое подмножество произведения компактов с топологией произведения . Как следствие теоремы Тихонова , это произведение, а следовательно, и единичный шар внутри, компактны.
Эта теорема имеет приложения в физике, когда описывается множество состояний алгебры наблюдаемых, а именно то, что любое состояние может быть записано как выпуклая линейная комбинация так называемых чистых состояний.
История
По словам Лоуренса Narici и Эдвард Бекенштейн, то Алаогл теоремы является «очень важным результатом - возможно самым важным факт о слабой * топологии - [что] отголоски по всему функциональному анализу.» [2] В 1912 году Хелли доказал, что единичный шар непрерывного сопряженного пространствасчетно слабо- * компактно. [3] В 1932 году Стефан Банах доказал, что замкнутый единичный шар в непрерывном сопряженном пространстве любого сепарабельного нормированного пространства секвенциально слабо- * компактен (Банах рассматривал только секвенциальную компактность ). [3] Доказательство для общего случая было опубликовано в 1940 году математиком Леонидасом Алаоглу . Согласно Pietsch [2007], есть по крайней мере 12 математиков, которые могут претендовать на эту теорему или на ее важного предшественника. [2]
Теорема Бурбаки – Алаоглу является обобщением [4] [5] исходной теоремы Бурбаки на двойственные топологии на локально выпуклых пространствах . Эта теорема также называется теоремой Банаха-Алаоглу или теоремой о слабой компактности, а обычно ее называют просто теоремой Алаоглу [2]
Заявление
Если векторное пространство над полем тогда будет обозначать алгебраическое двойственное пространство ки эти два пространства впредь ассоциируются с билинейным оценочным отображением определяется
где тройка образует дуальную систему, называемую канонической дуальной системой .
Если является топологическим векторным пространством (TVS), то его непрерывное сопряженное пространство обозначим через где всегда держит. Обозначим слабую * топологию на от и обозначим слабую * топологию на от Слабая * топология также называется топологией поточечной сходимости, поскольку дано отображениеи сеть карт сеть сходится к в этой топологии тогда и только тогда, когда для каждой точки в домене сеть ценностей сходится к значению
Теорема Алаоглу [3] - Для любого топологического векторного пространства (TVS)( не обязательно хаусдорфово или локально выпуклое ) с непрерывным сопряженным пространством полярная
любого района происхождения в компактна в слабой * топологии [примечание 1] на Более того, равен полюсу относительно канонической системы и это также компактное подмножество
Доказательство с использованием теории двойственности
Обозначим нижележащим полем от что либо действительные числа или комплексные числа В этом доказательстве будут использоваться некоторые из основных свойств, перечисленных в статьях: полярное множество , двойственная система и непрерывный линейный оператор .
Чтобы начать доказательство, напомним некоторые определения и легко проверяемые результаты. Когданаделен слабой * топологией то это хаусдорфово локально выпуклое топологическое векторное пространство обозначается через Космос всегда полная ТВС ; тем не мение, может не быть полным пространством, поэтому это доказательство включает в себя пространство В частности, это доказательство будет использовать тот факт, что подмножество полного хаусдорфова пространства компактно, если (и только если) оно замкнуто и вполне ограничено . Важно отметить, что топология подпространств , наследуется от равно В этом легко убедиться, показав, что при любом сетка в сходится к в одной из этих топологий тогда и только тогда, когда она также сходится к в другой топологии (вывод следует из того, что две топологии равны тогда и только тогда, когда они имеют точно такие же сходящиеся сети).
Тройка это двойная пара, хотя в отличие откак правило, двойная система не гарантируется. На всем протяжении, если не указано иное, все полярные множества будут взяты относительно канонического спаривания.
Позволять быть окрестностью начала координат в и разреши:
- быть полярником относительно канонического спаривания ;
- быть биполярным U {\ displaystyle U} относительно ;
- быть полярником относительно канонической дуальной системы
Хорошо известный факт о полярах множеств состоит в том, что
(1) Покажите, что это -замкнутое подмножество Позволять и предположим, что это сеть в что сходится к в Сделать вывод, что достаточно (и необходимо) показать, что для каждого Так как в скалярном поле и каждое значение принадлежит закрытому (в ) подмножество так же должен лимит этой сети принадлежат к этому набору. Таким образом
(2) Покажите, что а затем заключаем, что является замкнутым подмножеством обоих а также Включение выполняется, потому что каждый непрерывный линейный функционал (в частности) является линейным функционалом. Для обратного включения позволять чтобы что в точности утверждает, что линейный функционал ограничен в окрестности ; таким образом- линейный непрерывный функционал (т. е.) и другие по желанию. Используя (1) и тот факт, что пересечение замкнуто в топологии подпространств на претензия о быть закрытым следует.
(3) Покажите, что это - вполне ограниченное подмножествоПо биполярной теореме , где, потому что окрестности является поглощая подмножество из то же самое должно быть верно и для набора ; можно доказать, что отсюда следует, что это - ограниченное подмножество из Так как различает точку из подмножество является -ограничен тогда и только тогда, когда он - полностью ограничен . Так, в частности, is also -totally bounded.
(4) Conclude that is also a -totally bounded subset of Recall that the topology on is identical to the subspace topology that inherits from This fact, together with (3) and the definition of "totally bounded", implies that is a -totally bounded subset of
(5) Finally, deduce that is a -compact subset of Because is a complete TVS and is a closed (by (2)) and totally bounded (by (4)) subset of it follows that is compact. ∎
If is a normed vector space, then the polar of a neighborhood is closed and norm-bounded in the dual space. In particular, if is the open (or closed) unit ball in then the polar of is the closed unit ball in the continuous dual space of (with the usual dual norm). Consequently, this theorem can be specialized to:
- Banach-Alaoglu theorem: If is a normed space then the closed unit ball in the continuous dual space (endowed with its usual operator norm) is compact with respect to the weak-* topology.
When the continuous dual space of is an infinite dimensional normed space then it is impossible for the closed unit ball in to be a compact subset when has its usual norm topology. This is because the unit ball in the norm topology is compact if and only if the space is finite-dimensional (cf. F. Riesz theorem). This theorem is one example of the utility of having different topologies on the same vector space.
It should be cautioned that despite appearances, the Banach–Alaoglu theorem does not imply that the weak-* topology is locally compact. This is because the closed unit ball is only a neighborhood of the origin in the strong topology, but is usually not a neighborhood of the origin in the weak-* topology, as it has empty interior in the weak* topology, unless the space is finite-dimensional. In fact, it is a result of Weil that all locally compact Hausdorff topological vector spaces must be finite-dimensional.
Elementary proof
The following proof involves only elementary concepts from topology, set theory, and functional analysis.
Denote by the underlying field of by which is either the real numbers or complex numbers For any real let
denote the closed ball of radius at the origin in which is a compact and closed subset of
Because is a neighborhood of the origin in it is also an absorbing subset of so for every there exists a real number such that Let
denote the polar of with respect to the canonical dual system As is now shown, this polar set is the same as the polar of with respect to
Proof that The inclusion holds because every continuous linear functional is (in particular) a linear functional. For the reverse inclusion let so that which states exactly that the linear functional is bounded on the neighborhood ; thus is a continuous linear functional (that is, ) and so as desired.
The rest of this proof requires a proper understanding how the Cartesian product is identified as the space of all functions of the form An explanation is now given for readers who are interested.
Premiere on identification of functions with tuples |
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The Cartesian product is usually thought of as the set of all -indexed tuples but, as is now described, it can also be identified with the space of all functions having prototype
This is the reason why many authors write, often without comment, the equality and why the Cartesian product is sometimes taken as the definition of the set of maps However, the Cartesian product, being the (categorical) product in the category of sets (which is a type of inverse limit), also comes equipped with associated maps that are known as its (coordinate) projections. The Cartesian product's canonical projection at any given is the function
where under the above identification, sends a function to In words, for a point and function "plugging into " is the same as "plugging into ".
The set is assumed to be endowed with the product topology. It is well known that the product topology is identical to the topology of pointwise convergence. This is because given and a net where and every is an element of then the net converges in the product topology if and only if
where and Thus converges to in the product topology if and only if it converges to pointwise on Also used in this proof will be the fact that the topology of pointwise convergence is preserved when passing to topological subspaces. This means, for example, that if for every is some (topological) subspace of then the topology of pointwise convergence (or equivalently, the product topology) on is equal to the subspace topology that the set inherits from |
Having established that [note 2] to reduce symbol clutter, this olar set will be denoted by
unless an attempt is being made to draw attention to the definition of or
The proof of the theorem will be complete once the following statements are verified:
- is a closed subset of
- Here is endowed with the topology of pointwise convergence, which is identical to the product topology.
- denotes the closed ball of radius centered at For each was defined at the start of this proof as any real that satisfies (so in particular, is a valid choice for each ).
These statements imply that is a closed subset of where this product space is compact by Tychonoff's theorem[note 3] (because every closed ball is a compact space). Because a closed subset of a compact space is compact, it follows that is compact, which is the main conclusion of the Banach–Alaoglu theorem.
Proof of (1):
The algebraic dual space is always a closed subset of (this is proved in the lemma below for readers who are not familiar with this result). To prove that is closed in it suffices to show that the set defined by
is a closed subset of because then is an intersection of two closed subsets of Let and suppose that is a net in that converges to in To conclude that it is sufficient (and necessary) to show that for every (or equivalently, that ). Because in the scalar field and every value belongs to the closed (in ) subset so too must this net's limit belong to this closed set. Thus which completes the proof of (1).
As a side note, this proof can be generalized to prove the following more general result, from which the above conclusion follows as the special case and
- Proposition: If is any set and if is a closed subset of a topological space then is a closed subset of with respect to the topology of pointwise convergence.
Proof of (2):
For any let denote the projection to the th coordinate (as defined above). To prove that it is sufficient (and necessary) to show that for every So fix and let ; it remains to show that The defining condition on was that which implies that Because the linear functional satisfies and so implies
Thus which shows that as desired.
The elementary proof above actually shows that if is any subset that satisfies (such as any absorbing subset of ), then is a weak-* compact subset of
As a side note, with the help of the above elementary proof, it may be shown (see this footnote)[note 4] that
where is defined by for every with (as in the proof) and
In fact,
- and
where denotes the intersection of all sets belonging to
This implies (among other things[note 5]) that the unique least element of with respect to ; this may be used as an alternative definition of this (necessarily convex and balanced) set. The function is a seminorm and it is unchanged if is replaced by the convex balanced hull of (because ). Similarly, because is also unchanged if is replaced by its closure in
Lemma — The algebraic dual space of any vector space over a field (where is or ) is a closed subset of in the topology of pointwise convergence. (The vector space need not be endowed with any topology).
Proof of lemma |
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A net in is by definition a function from a non-empty directed set Every sequence in which by definition is just a function of the form is also a net. As with sequences, the value of a net at an index is denoted by ; however, for this proof, this value may also be denoted by the usual function parentheses notation Similarly for function composition, if is any function then the net (or sequence) that results from "plugging into " is just the function although this is typically denoted by (or by if is a sequence). In this proof, this resulting net may be denoted by any of the following notations depending on whichever notation is cleanest or most clearly communicates the intended information. In particular, if is continuous and in then the conclusion commonly written as may instead be written as or Start of proof: Let and suppose that is a net in the converges to in If then will denote 's net of values at To conclude that it must be shown that is a linear functional so let be a scalar and let The topology on is the topology of pointwise convergence so by considering the points and the convergence of in implies that each of the following nets of scalars converges in
which proves that Because also and limits in are unique, it follows that as desired.
which proves that Because also it follows that as desired. |
Corollary to lemma — When the algebraic dual space of a vector space is equipped with the topology of pointwise convergence (also known as the weak-* topology) then the resulting topological vector space (TVS) is a complete Hausdorff locally convex TVS.
Proof of corollary |
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Because the underlying field is a complete Hausdorff locally convex TVS, the same is true of the Cartesian product A closed subset of a complete space is complete, so by the lemma, the space is complete. |
Последовательная теорема Банаха – Алаоглу.
A special case of the Banach–Alaoglu theorem is the sequential version of the theorem, which asserts that the closed unit ball of the dual space of a separable normed vector space is sequentially compact in the weak-* topology. In fact, the weak* topology on the closed unit ball of the dual of a separable space is metrizable, and thus compactness and sequential compactness are equivalent.
Specifically, let be a separable normed space and the closed unit ball in Since is separable, let be a countable dense subset. Then the following defines a metric, where for any
in which denotes the duality pairing of with Sequential compactness of in this metric can be shown by a diagonalization argument similar to the one employed in the proof of the Arzelà–Ascoli theorem.
Due to the constructive nature of its proof (as opposed to the general case, which is based on the axiom of choice), the sequential Banach–Alaoglu theorem is often used in the field of partial differential equations to construct solutions to PDE or variational problems. For instance, if one wants to minimize a functional on the dual of a separable normed vector space one common strategy is to first construct a minimizing sequence which approaches the infimum of use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit and then establish that is a minimizer of The last step often requires to obey a (sequential) lower semi-continuity property in the weak* topology.
When is the space of finite Radon measures on the real line (so that is the space of continuous functions vanishing at infinity, by the Riesz representation theorem), the sequential Banach–Alaoglu theorem is equivalent to the Helly selection theorem.
For every let
and
Because each is a compact subset of the complex plane, is also compact in the product topology by Tychonoff's theorem.
The closed unit ball in can be identified as a subset of in a natural way:
This map is injective and continuous, with having the weak-* topology and the product topology. This map's inverse, defined on its range, is also continuous.
To finish proving this theorem, it will now be shown that the range of the above map is closed. Given a net
in the functional defined by
lies in
Последствия
- Consequences for normed spaces
Assume that is a normed space and endow its continuous dual space with the usual dual norm.
- The closed unit ball in is weak-* compact.[3] So if is infinite dimensional then its closed unit ball is necessarily not compact in the norm topology by the F. Riesz theorem (despite it being weak-* compact).
- A Banach space is reflexive if and only if its closed unit ball is -compact.[3]
- If is a reflexive Banach space, then every bounded sequence in has a weakly convergent subsequence. (This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of ; or, more succinctly, by applying the Eberlein–Šmulian theorem.) For example, suppose that is the space L p ( μ ) {\displaystyle L^{p}(\mu )}
, Let be a bounded sequence of functions in Then there exists a subsequence and an such that
- Consequences for Hilbert spaces
- In a Hilbert space, every bounded and closed set is weakly relatively compact, hence every bounded net has a weakly convergent subnet (Hilbert spaces are reflexive).
- As norm-closed, convex sets are weakly closed (Hahn–Banach theorem), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact.
- Closed and bounded sets in are precompact with respect to the weak operator topology (the weak operator topology is weaker than the ultraweak topology which is in turn the weak-* topology with respect to the predual of the trace class operators). Hence bounded sequences of operators have a weak accumulation point. As a consequence, has the Heine–Borel property, if equipped with either the weak operator or the ultraweak topology.
Отношение к аксиоме выбора
Since the Banach–Alaoglu theorem is usually proven via Tychonoff's theorem, it relies on the ZFC axiomatic framework, and in particular the axiom of choice. Most mainstream functional analysis also relies on ZFC. However, the theorem does not rely upon the axiom of choice in the separable case (see above): in this case one actually has a constructive proof. In the non-separable case, the ultrafilter Lemma, which is strictly weaker than the axiom of choice, suffices for the proof of the Banach-Alaoglu theorem, and is in fact equivalent to it.
Смотрите также
- Bishop–Phelps theorem
- Banach–Mazur theorem
- Delta-compactness theorem
- Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
- Goldstine theorem
- James' theorem
- Krein-Milman theorem
- Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space
- Topological vector space – Vector space with a notion of nearness
Заметки
- ^ Explicitly, a subset is said to be "compact (resp. totally bounded, etc.) in the weak-* topology" if when is given the weak-* topology and the subset is given the subspace topology inherited from then is a compact (resp. totally bounded, etc.) space.
- ^ If denotes the topology that is (originally) endowed with, then the equality shows that the polar of is dependent only on (and ) and that the rest of the topology can be ignored. To clarify what is meant, suppose is any TVS topology on such that the set is (also) a neighborhood of the origin in Denote the continuous dual space of by and denote the polar of with respect to by
- ^ Because every is also a Hausdorff space, the conclusion that is compact only requires the so-called "Tychonoff's theorem for compact Hausdorff spaces," which is equivalent to the ultrafilter lemma and strictly weaker than the axiom of choice.
- ^ For any non-empty subset the equality holds (the intersection on the left is a closed, rather than open, disk − possibly of radius − because it is an intersection of closed subsets of and so must itself be closed). For every let so that the previous set equality implies From it follows that and thereby making the least element of with respect to (In fact, the family is closed under (non-nullary) arbitrary intersections and also under finite unions of at least one set). Statement (2) in the above elementary proof showed that and are not empty and moreover, it also even showed that has an element that satisfies for every which implies that for every The inclusion is immediate so to prove the reverse inclusion, let By definition, if and only if so let and it remains to show that From it follows that which implies that as desired.
- ^ This tuple is the least element of with respect to natural induced pointwise partial order defined by if and only if for every Thus, every neighborhood of the origin in can be associated with this unique (minimum) function For any if is such that then so that in particular, and for every
Рекомендации
- ^ Rudin 1991, Theorem 3.15.
- ^ a b c Narici & Beckenstein 2011, pp. 235-240.
- ^ a b c d e Narici & Beckenstein 2011, pp. 225-273.
- ^ Köthe 1969, Theorem (4) in §20.9.
- ^ Meise & Vogt 1997, Theorem 23.5.
- Köthe, Gottfried (1969). Topological Vector Spaces I. New York: Springer-Verlag. See §20.9.
- Meise, Reinhold; Vogt, Dietmar (1997). Introduction to Functional Analysis. Oxford: Clarendon Press. ISBN 0-19-851485-9. See Theorem 23.5, p. 264.
- 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 (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. See Theorem 3.15, p. 68.
- 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 (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
дальнейшее чтение
- John B. Conway (1994). A course in functional analysis (2nd ed.). Berlin: Springer-Verlag. ISBN 0-387-97245-5. See Chapter 5, section 3.
- Peter B. Lax (2002). Functional Analysis. Wiley-Interscience. pp. 120–121. ISBN 0-471-55604-1.