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- mathematischer Satz (de)
- theorem (en)
- théorème mathématique (fr)
- teorema matematico (it)
- teorema matemàtic (ca)
- matematični izrek (sl)
- 賦範向量空間的對偶中，閉單位球是弱*拓撲的緊集 (zh)
- twierdzenie analizy funkcjonalnej o kulach w przestrzeniach unormowanych (pl)
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- If is a normed space then the closed unit ball in the continuous dual space is compact with respect to the weak-* topology. (en)
- Let be a subset of a vector space over the field and for every real number endow the closed ball with its usual topology .
Define
If for every is a real number such that then is a closed and compact subspace of the product space . (en)
- The algebraic dual space of any vector space over a field is a closed subset of in the topology of pointwise convergence. . (en)
- Let be a normed space and let denote the closed unit ball of its continuous dual space Then has the following property, which is called or : whenever is a cover of by weak-* closed subsets of such that has the finite intersection property, then is not empty. (en)
- For any topological vector space with continuous dual space the polar
of any neighborhood of origin in is compact in the weak-* topology on Moreover, is equal to the polar of with respect to the canonical system and it is also a compact subset of (en)
- When the algebraic dual space of a vector space is equipped with the topology of pointwise convergence then the resulting topological space is a complete Hausdorff locally convex topological vector space. (en)
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- Proposition (en)
- Lemma (en)
- Alaoglu theorem (en)
- Banach–Alaoglu theorem (en)
- Corollary to lemma (en)
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- Denote by the underlying field of by which is either the real numbers or complex numbers
This proof will use some of the basic properties that are listed in the articles: polar set, dual system, and continuous linear operator.
To start the proof, some definitions and readily verified results are recalled. When is endowed with the weak-* topology then this Hausdorff locally convex topological vector space is denoted by
The space is always a complete TVS; however, may fail to be a complete space, which is the reason why this proof involves the space
Specifically, this proof will use the fact that a subset of a complete Hausdorff space is compact if it is closed and totally bounded.
Importantly, the subspace topology that inherits from is equal to This can be readily verified by showing that given any a net in converges to in one of these topologies if and only if it also converges to in the other topology .
The triple is a dual pairing although unlike it is in general not guaranteed to be a dual system.
Throughout, unless stated otherwise, all polar sets will be taken with respect to the canonical pairing
Let be a neighborhood of the origin in and let:
* be the polar of with respect to the canonical pairing ;
* be the bipolar of with respect to ;
* be the polar of with respect to the canonical dual system Note that
A well known fact about polar sets is that
# Show that is a -closed subset of Let and suppose that is a net in that converges to in To conclude that it is sufficient to show that for every Because in the scalar field and every value belongs to the closed subset so too must this net's limit belong to this set. Thus
# Show that and then conclude that is a closed subset of both and The inclusion holds because every continuous linear functional is 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 and so as desired. Using and the fact that the intersection is closed in the subspace topology on the claim about being closed follows.
# Show that is a -totally bounded subset of By the bipolar theorem, where because the neighborhood is an absorbing subset of the same must be true of the set it is possible to prove that this implies that is a -bounded subset of Because distinguishes points of a subset of is -bounded if and only if it is -totally bounded. So in particular, is also -totally bounded.
# 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 and the definition of "totally bounded", implies that is a -totally bounded subset of
# Finally, deduce that is a -compact subset of Because is a complete TVS and is a closed and totally bounded subset of it follows that is compact. (en)
- For every let
and let
be endowed with the product topology.
Because every is a compact subset of the complex plane, Tychonoff's theorem guarantees that their product is compact.
The closed unit ball in denoted by can be identified as a subset of in a natural way:
This map is injective and it is continuous when has the weak-* topology.
This map's inverse, defined on its image, is also continuous.
It will now be shown that the image of the above map is closed, which will complete the proof of the theorem.
Given a point and a net in the image of indexed by such that
the functional defined by
lies in and (en)
- Assume that is a topological vector space with continuous dual space and that is a neighborhood of the origin.
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
Thus the hypotheses of the above proposition are satisfied, and so the set is therefore compact in the weak-* topology.
The proof of the Banach–Alaoglu theorem will be complete once it is shown that
where recall that was defined as
Proof that
Because the conclusion is equivalent to
If then which states exactly that the linear functional is bounded on the neighborhood thus is a continuous linear functional , as desired. (en)
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- Primer on product/function spaces, nets, and pointwise convergence (en)
- Proof that Banach–Alaoglu follows from the proposition above (en)
- Proof of corollary to lemma (en)
- Proof of lemma (en)
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- Banach–Alaoglu theorem (en)
- Teorema de Banach-Alaoglu (ca)
- Teorema de Banach-Alaoglu (es)
- Satz von Banach-Alaoglu (de)
- Teorema di Banach-Alaoglu (it)
- Théorème de Banach-Alaoglu-Bourbaki (fr)
- バナッハ＝アラオグルの定理 (ja)
- 바나흐-앨러오글루 정리 (ko)
- Twierdzenie Banacha-Alaoglu (pl)
- Stelling van Banach-Alaoglu (nl)
- Banach–Alaoglus sats (sv)
- Теорема Алаоглу (ru)
- 巴拿赫-阿勞格魯定理 (zh)
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