{"history":[],"article":{"id":"c3f77f54-0638-41d7-b20c-19c116ec28e6","title":"Banach–Alaoglu theorem","slug":"banach-alaoglu-theorem","content":":::summary\n:::\n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and related branches of [mathematics](https://bliptext.com/articles/mathematics), the **Banach–Alaoglu theorem** (also known as **Alaoglu's theorem**) states that the [closed](https://bliptext.com/articles/closed-set) [unit ball](https://bliptext.com/articles/ball-mathematics) of the [dual space](https://bliptext.com/articles/dual-space) of a [normed vector space](https://bliptext.com/articles/normed-vector-space) is [compact](https://bliptext.com/articles/compact-space) in the [weak* topology](https://bliptext.com/articles/weak-topology).\nA common proof identifies the unit ball with the weak-* topology as a closed subset of a [product](https://bliptext.com/articles/cartesian-product) of compact sets with the [product](https://bliptext.com/articles/cartesian-product) topology.\nAs a consequence of [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem), this product, and hence the unit ball within, is compact.\nThis theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states.\n\n# History\nAccording to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe most important fact about the [weak-* topology](https://bliptext.com/articles/weak-topology)—[that] echos throughout functional analysis.”\nIn 1912, Helly proved that the unit ball of the continuous dual space of C([a, b]) is countably weak-* compact.\nIn 1932, [Stefan Banach](https://bliptext.com/articles/stefan-banach) proved that the closed unit ball in the continuous dual space of any [separable](https://bliptext.com/articles/separable-metric-space) [normed space](https://bliptext.com/articles/normed-space) is sequentially weak-* compact (Banach only considered [sequential compactness](https://bliptext.com/articles/sequential-compactness)).\nThe proof for the general case was published in 1940 by the mathematician [Leonidas Alaoglu](https://bliptext.com/articles/leonidas-alaoglu).\nAccording to Pietsch [2007], there are at least twelve mathematicians who can lay claim to this theorem or an important predecessor to it.\nThe **[Bourbaki](https://bliptext.com/articles/nicolas-bourbaki)–Alaoglu theorem** is a generalization of the original theorem by [Bourbaki](https://bliptext.com/articles/nicolas-bourbaki) to [dual topologies](https://bliptext.com/articles/dual-topology) on [locally convex spaces](https://bliptext.com/articles/locally-convex-space).\nThis theorem is also called the **Banach–**Alaoglu theorem**** or the **weak-* compactness theorem** and it is commonly called simply the **Alaoglu theorem**.\n\n# Statement\nIf X is a vector space over the field \\mathbb{K} then X^{\\#} will denote the [algebraic dual space](https://bliptext.com/articles/algebraic-dual-space) of X and these two spaces are henceforth associated with the [bilinear](https://bliptext.com/articles/bilinear-map) defined by\nwhere the triple forms a [dual system](https://bliptext.com/articles/dual-system) called the.\nIf X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) then its [continuous dual space](https://bliptext.com/articles/continuous-dual-space) will be denoted by X^{\\prime}, where always holds.\nDenote the [weak-* topology](https://bliptext.com/articles/weak-topology) on X^{\\#} by and denote the [weak-* topology](https://bliptext.com/articles/weak-topology) on X^{\\prime} by The [weak-* topology](https://bliptext.com/articles/weak-topology) is also called the [topology of pointwise convergence](https://bliptext.com/articles/topology-of-pointwise-convergence) because given a map f and a [net](https://bliptext.com/articles/net-mathematics) of maps the [net](https://bliptext.com/articles/net-mathematics) f_{\\bull} converges to f in this topology if and only if for every point x in the domain, the [net](https://bliptext.com/articles/net-mathematics) of values converges to the value f(x).\n\n## Proof involving duality theory\nIf X is a [normed vector space](https://bliptext.com/articles/normed-vector-space), then the polar of a neighborhood is closed and norm-bounded in the dual space.\nIn particular, if U is the open (or closed) unit ball in X then the polar of U is the closed unit ball in the continuous dual space X^{\\prime} of X (with the [usual dual norm](https://bliptext.com/articles/operator-norm)).\nConsequently, this theorem can be specialized to:\nWhen the continuous dual space X^{\\prime} of X is an infinite dimensional normed space then it is for the closed unit ball in X^{\\prime} to be a compact subset when X^{\\prime} has its usual norm topology.\nThis is because the unit ball in the norm topology is compact if and only if the space is finite-dimensional (cf. [F. Riesz theorem](https://bliptext.com/articles/f-riesz-theorem)).\nThis theorem is one example of the utility of having different topologies on the same vector space.\nIt should be cautioned that despite appearances, the Banach–Alaoglu theorem does imply that the weak-* topology is [locally compact](https://bliptext.com/articles/locally-compact).\nThis is because the closed unit ball is only a neighborhood of the origin in the [strong topology](https://bliptext.com/articles/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.\nIn fact, it is a result of [Weil](https://bliptext.com/articles/andr-weil) that all [locally compact](https://bliptext.com/articles/locally-compact) [Hausdorff](https://bliptext.com/articles/hausdorff-space) topological vector spaces must be finite-dimensional.\n\n## Elementary proof\nThe following [elementary proof](https://bliptext.com/articles/elementary-proof) does not utilize [duality theory](https://bliptext.com/articles/duality-theory) and requires only basic concepts from set theory, topology, and functional analysis.\nWhat is needed from topology is a working knowledge of [net](https://bliptext.com/articles/net-mathematics) [convergence](https://bliptext.com/articles/net-convergence) in [topological spaces](https://bliptext.com/articles/topological-space) and familiarity with the fact that a [linear functional is continuous](https://bliptext.com/articles/continuous-linear-functional) if and only if it is [bounded on a neighborhood](https://bliptext.com/articles/continuous-linear-operator) of the origin (see the articles on [continuous linear functionals](https://bliptext.com/articles/continuous-linear-functional) and [sublinear functionals](https://bliptext.com/articles/sublinear-functional) for details).\nAlso required is a proper understanding of the technical details of how the space of all functions of the form is identified as the [Cartesian product](https://bliptext.com/articles/cartesian-product) and the relationship between [pointwise convergence](https://bliptext.com/articles/pointwise-convergence), the [product topology](https://bliptext.com/articles/product-topology), and [subspace topologies](https://bliptext.com/articles/subspace-topology) they induce on subsets such as the [algebraic dual space](https://bliptext.com/articles/algebraic-dual-space) X^{\\#} and products of subspaces such as An explanation of these details is now given for readers who are interested.\nFor every real r, will denote the closed ball of radius r centered at 0 and for any\n**Identification of functions with tuples**\nThe Cartesian product is usually thought of as the set of all X-indexed [tuples](https://bliptext.com/articles/tuple) but, since [tuples](https://bliptext.com/articles/tuple) are technically just functions from an indexing set, it can also be identified with the space of all functions having prototype as is now described:\nThis is the reason why many authors write, often without comment, the equality\nand why the Cartesian product is sometimes taken as the definition of the set of maps (or conversely).\nHowever, the Cartesian product, being the [(categorical) product](https://bliptext.com/articles/product-[category](https://bliptext.com/articles/category-mathematics)-theory) in the [category](https://bliptext.com/articles/category-mathematics) of [sets](https://bliptext.com/articles/category-of-sets) (which is a type of [inverse limit](https://bliptext.com/articles/inverse-limit)), also comes equipped with associated maps that are known as its (coordinate).\nThe at a given point z \\in X is the function\nwhere under the above identification, \\Pr{}_z sends a function to Stated in words, for a point z and function s, \"plugging z into s\" is the same as \"plugging s into \\Pr{}_z\".\nIn particular, suppose that are non-negative real numbers.\nThen where under the above identification of tuples with functions, is the set of all functions such that for every x \\in X.\nIf a subset [partitions](https://bliptext.com/articles/partition-of-a-set) X into then the linear bijection\ncanonically identifies these two Cartesian products; moreover, this map is a [homeomorphism](https://bliptext.com/articles/homeomorphism) when these products are endowed with their product topologies.\nIn terms of function spaces, this bijection could be expressed as\n**Notation for nets and function composition with nets**\nA [net](https://bliptext.com/articles/net-mathematics) in X is by definition a function from a non-empty [directed set](https://bliptext.com/articles/directed-set) (I, \\leq).\nEvery [sequence](https://bliptext.com/articles/sequence) in X, which by definition is just a function of the form \\N \\to X, is also a net.\nAs with sequences, the value of a net x_{\\bull} at an index i \\in I is denoted by x_i; however, for this proof, this value x_i may also be denoted by the usual function parentheses notation\nSimilarly for [function composition](https://bliptext.com/articles/function-composition), if F : X \\to Y is any function then the net (or sequence) that results from \"plugging x_{\\bull} into F\" is just the function although this is typically denoted by (or by if x_{\\bull} is a sequence).\nIn the proofs below, this resulting net may be denoted by any of the following notations\ndepending on whichever notation is cleanest or most clearly communicates the intended information.\nIn particular, if F : X \\to Y is continuous and in X, then the conclusion commonly written as may instead be written as or\n**Topology**\nThe set is assumed to be endowed with the [product topology](https://bliptext.com/articles/product-topology).\nIt is well known that the product topology is identical to the [topology of pointwise convergence](https://bliptext.com/articles/topology-of-pointwise-convergence).\nThis is because given f and a [net](https://bliptext.com/articles/net-mathematics) where f and every f_i is an element of then the [net](https://bliptext.com/articles/net-mathematics) [converges](https://bliptext.com/articles/convergent-net) in the product topology if and only if\nwhere because and\nthis happens if and only if\nThus converges to f in the product topology if and only if it converges to f pointwise on X.\nThis proof will also use the fact that the topology of pointwise convergence is preserved when passing to [topological subspaces](https://bliptext.com/articles/topological-subspace).\nThis means, for example, that if for every x \\in X, is some [(topological) subspace](https://bliptext.com/articles/topological-subspace) of \\mathbb{K} then the topology of pointwise convergence (or equivalently, the product topology) on is equal to the [subspace topology](https://bliptext.com/articles/subspace-topology) that the set inherits from\nAnd if S_x is closed in \\mathbb{K} for every x \\in X, then is a closed subset of\n**Characterization of **\nAn important fact used by the proof is that for any real r,\nwhere \\,\\sup\\, denotes the [supremum](https://bliptext.com/articles/supremum) and\nAs a side note, this characterization does not hold if the closed ball B_r is replaced with the open ball (and replacing with the strict inequality will not change this; for counter-examples, consider and the [identity map](https://bliptext.com/articles/identity-map) on X).\nThe essence of the Banach–Alaoglu theorem can be found in the next proposition, from which the Banach–Alaoglu theorem follows.\nUnlike the Banach–Alaoglu theorem, this proposition does require the vector space X to endowed with any topology.\nBefore proving the proposition above, it is first shown how the Banach–Alaoglu theorem follows from it (unlike the proposition, Banach–Alaoglu assumes that X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) and that U is a neighborhood of the origin).\nThe conclusion that the set is closed can also be reached by applying the following more general result, this time proved using nets, to the special case and B := B_1.\nLet and suppose that is a net in X^{\\#} the converges to f in To conclude that it must be shown that f is a linear functional.\nSo let s be a scalar and let x, y \\in X.\nFor any z \\in X, let denote\nBecause in which has the topology of pointwise convergence, in \\mathbb{K} for every z \\in X. By using in place of z, it follows that each of the following nets of scalars converges in \\mathbb{K}:\nProof that\nLet be the \"multiplication by s\" map defined by Because M is continuous and in \\mathbb{K}, it follows that where the right hand side is and the left hand side is\nwhich proves that Because also and limits in \\mathbb{K} are unique, it follows that as desired.\nProof that\nDefine a net by letting for every i \\in I. Because and it follows that in Let be the addition map defined by The continuity of A implies that in \\mathbb{K} where the right hand side is and the left hand side is\nwhich proves that Because also it follows that as desired.\n[The lemma above](https://bliptext.com/articles/) actually also follows from its corollary below since is a Hausdorff [complete uniform space](https://bliptext.com/articles/complete-uniform-space) and any subset of such a space (in particular X^{\\#}) is closed if and only if it is complete.\nBecause the underlying field \\mathbb{K} is a complete Hausdorff locally convex topological vector space, the same is true of the [product space](https://bliptext.com/articles/product-space)\nA closed subset of a complete space is complete, so by the lemma, the space is complete.\nThe above elementary proof of the Banach–Alaoglu theorem actually shows that if is any subset that satisfies (such as any [absorbing subset](https://bliptext.com/articles/absorbing-set) of X), then is a [weak-* compact](https://bliptext.com/articles/weak-topology) subset of X^{\\#}.\nAs a side note, with the help of the above elementary proof, it may be shown (see this footnote)\nthat there exist X-indexed non-negative real numbers such that\nwhere these real numbers m_{\\bull} can also be chosen to be \"minimal\" in the following sense:\nusing (so P = U^{\\#} as in the proof) and defining the notation for any if then and for every x \\in X, which shows that these numbers m_{\\bull} are unique; indeed, this [infimum](https://bliptext.com/articles/infimum) formula can be used to define them.\nIn fact, if denotes the set of all such products of closed balls containing the polar set P, then\nwhere denotes the intersection of all sets belonging to\nThis implies (among other things )\nthat the unique [least element](https://bliptext.com/articles/least-element) of with respect to this may be used as an alternative definition of this (necessarily [convex](https://bliptext.com/articles/convex-set) and [balanced](https://bliptext.com/articles/balanced-set)) set.\nThe function is a [seminorm](https://bliptext.com/articles/seminorm) and it is unchanged if U is replaced by the [convex balanced hull](https://bliptext.com/articles/convex-balanced-hull) of U (because ).\nSimilarly, because m_{\\bull} is also unchanged if U is replaced by its [closure](https://bliptext.com/articles/closure-topology) in X.\n\n# Sequential Banach–Alaoglu theorem\nA 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](https://bliptext.com/articles/separable-metric-space) normed vector space is [sequentially compact](https://bliptext.com/articles/sequentially-compact) in the weak-* topology.\nIn fact, the weak* topology on the closed unit ball of the dual of a separable space is [metrizable](https://bliptext.com/articles/metrizable), and thus compactness and sequential compactness are equivalent.\nSpecifically, let X be a separable normed space and B the closed unit ball in X^{\\prime}.\nSince X is separable, let be a countable dense subset.\nThen the following defines a metric, where for any x, y \\in B\nin which denotes the duality pairing of X^{\\prime} with X. Sequential compactness of B in this metric can be shown by a [diagonalization argument](https://bliptext.com/articles/diagonalization-argument) similar to the one employed in the proof of the [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem).\nDue 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](https://bliptext.com/articles/partial-differential-equations) to construct solutions to PDE or [variational problems](https://bliptext.com/articles/calculus-of-variations).\nFor instance, if one wants to minimize a functional on the dual of a separable normed vector space X, one common strategy is to first construct a minimizing sequence which approaches the infimum of F, use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit x, and then establish that x is a minimizer of F. The last step often requires F to obey a (sequential) [lower semi-continuity](https://bliptext.com/articles/semi-continuity) property in the weak* topology.\nWhen X^{\\prime} is the space of finite Radon measures on the real line (so that X = C_0(\\R) is the space of continuous functions vanishing at infinity, by the [Riesz representation theorem](https://bliptext.com/articles/riesz-markov-kakutani-representation-theorem)), the sequential Banach–Alaoglu theorem is equivalent to the [Helly selection theorem](https://bliptext.com/articles/helly-selection-theorem).\n\n# Consequences\n\n## Consequences for normed spaces\nAssume that X is a [normed space](https://bliptext.com/articles/normed-space) and endow its continuous dual space X^{\\prime} with the usual [dual norm](https://bliptext.com/articles/dual-norm).\n

The closed unit ball in X^{\\prime} is weak-* compact.\nSo if X^{\\prime} is infinite dimensional then its closed unit ball is necessarily compact in the norm topology by [F. Riesz's theorem](https://bliptext.com/articles/f-riesz-s-theorem) (despite it being weak-* compact).\n
A [Banach space](https://bliptext.com/articles/banach-space) is [reflexive](https://bliptext.com/articles/reflexive-space) if and only if its closed unit ball is -compact; this is known as [James' theorem](https://bliptext.com/articles/james-theorem).
If X is a [reflexive Banach space](https://bliptext.com/articles/reflexive-banach-space), then every bounded sequence in X has a weakly convergent subsequence.\n(This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of X; or, more succinctly, by applying the [Eberlein–Šmulian theorem](https://bliptext.com/articles/eberlein-mulian-theorem).)\nFor example, suppose that X is the space [Lp space](https://bliptext.com/articles/lp-space) L^p(\\mu) where and let q satisfy\nLet be a bounded sequence of functions in X. Then there exists a subsequence and an f \\in X such that\nThe corresponding result for p = 1 is not true, as L^1(\\mu) is not reflexive.

\n\n## Consequences for Hilbert spaces\n

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](https://bliptext.com/articles/reflexive-space)).
As norm-closed, convex sets are weakly closed ([Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem)), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact.
Closed and bounded sets in B(H) are precompact with respect to the [weak operator topology](https://bliptext.com/articles/weak-operator-topology) (the [weak operator topology](https://bliptext.com/articles/weak-operator-topology) is weaker than the [ultraweak topology](https://bliptext.com/articles/ultraweak-topology) which is in turn the weak-* topology with respect to the predual of B(H), the [trace class](https://bliptext.com/articles/trace-class) operators).\nHence bounded sequences of operators have a weak accumulation point.\nAs a consequence, B(H) has the [Heine–Borel property](https://bliptext.com/articles/heine-borel-theorem), if equipped with either the weak operator or the ultraweak topology.

\n\n# Relation to the axiom of choice and other statements\nThe Banach–Alaoglu may be proven by using [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem), which under the [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory) (**ZF**) axiomatic framework is equivalent to the [axiom of choice](https://bliptext.com/articles/axiom-of-choice).\nMost mainstream functional analysis relies on **ZF** + the axiom of choice, which is often denoted by **ZF**C.\nHowever, the theorem does rely upon the axiom of choice in the [separable](https://bliptext.com/articles/separable-space) case (see [above](https://bliptext.com/articles/)): in this case there actually exists a constructive proof.\nIn the general case of an arbitrary normed space, the [ultrafilter Lemma](https://bliptext.com/articles/ultrafilter-lemma), which is strictly weaker than the axiom of choice and equivalent to Tychonoff's theorem for compact spaces, suffices for the proof of the Banach–Alaoglu theorem, and is in fact equivalent to it.\nThe Banach–Alaoglu theorem is equivalent to the [ultrafilter lemma](https://bliptext.com/articles/ultrafilter-lemma), which implies the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) for [real vector spaces](https://bliptext.com/articles/real-vector-space) (****HB****) but is not equivalent to it (said differently, Banach–Alaoglu is also strictly stronger than ****HB****).\nHowever, the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) is equivalent to the following weak version of the Banach–Alaoglu theorem for normed space in which the conclusion of compactness (in the [weak-* topology](https://bliptext.com/articles/weak-topology) of the closed unit ball of the dual space) is replaced with the conclusion of (also sometimes called );\n[Compactness](https://bliptext.com/articles/compact-space) implies [convex compactness](https://bliptext.com/articles/) because a topological space is compact if and only if every [family](https://bliptext.com/articles/family-of-sets) of closed subsets having the [finite intersection property](https://bliptext.com/articles/finite-intersection-property) (FIP) has non-empty intersection.\nThe [definition of convex compactness](https://bliptext.com/articles/) is similar to this characterization of [compact spaces](https://bliptext.com/articles/compact-space) in terms of the FIP, except that it only involves those closed subsets that are also [convex](https://bliptext.com/articles/convex-set) (rather than all closed subsets).\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T20:03:57.159Z","updated_at":"2025-01-25T20:03:57.159Z","search_vector":"'alaoglu':2 'banach':1 'theorem':3","search_content":" \n \n\n# \nIn [functional analysis](https://bliptext.com/articles/functional-analysis) and related branches of [mathematics](https://bliptext.com/articles/mathematics), the **Banach–Alaoglu theorem** (also known as **Alaoglu's theorem**) states that the [closed](https://bliptext.com/articles/closed-set) [unit ball](https://bliptext.com/articles/ball-mathematics) of the [dual space](https://bliptext.com/articles/dual-space) of a [normed vector space](https://bliptext.com/articles/normed-vector-space) is [compact](https://bliptext.com/articles/compact-space) in the [weak* topology](https://bliptext.com/articles/weak-topology).\nA common proof identifies the unit ball with the weak-* topology as a closed subset of a [product](https://bliptext.com/articles/cartesian-product) of compact sets with the [product](https://bliptext.com/articles/cartesian-product) topology.\nAs a consequence of [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem), this product, and hence the unit ball within, is compact.\nThis theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states.\n\n# History\nAccording to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe most important fact about the [weak-* topology](https://bliptext.com/articles/weak-topology)—[that] echos throughout functional analysis.”\nIn 1912, Helly proved that the unit ball of the continuous dual space of C([a, b]) is countably weak-* compact.\nIn 1932, [Stefan Banach](https://bliptext.com/articles/stefan-banach) proved that the closed unit ball in the continuous dual space of any [separable](https://bliptext.com/articles/separable-metric-space) [normed space](https://bliptext.com/articles/normed-space) is sequentially weak-* compact (Banach only considered [sequential compactness](https://bliptext.com/articles/sequential-compactness)).\nThe proof for the general case was published in 1940 by the mathematician [Leonidas Alaoglu](https://bliptext.com/articles/leonidas-alaoglu).\nAccording to Pietsch [2007], there are at least twelve mathematicians who can lay claim to this theorem or an important predecessor to it.\nThe **[Bourbaki](https://bliptext.com/articles/nicolas-bourbaki)–Alaoglu theorem** is a generalization of the original theorem by [Bourbaki](https://bliptext.com/articles/nicolas-bourbaki) to [dual topologies](https://bliptext.com/articles/dual-topology) on [locally convex spaces](https://bliptext.com/articles/locally-convex-space).\nThis theorem is also called the **Banach–**Alaoglu theorem**** or the **weak-* compactness theorem** and it is commonly called simply the **Alaoglu theorem**.\n\n# Statement\nIf X is a vector space over the field \\mathbb{K} then X^{\\#} will denote the [algebraic dual space](https://bliptext.com/articles/algebraic-dual-space) of X and these two spaces are henceforth associated with the [bilinear](https://bliptext.com/articles/bilinear-map) defined by\nwhere the triple forms a [dual system](https://bliptext.com/articles/dual-system) called the.\nIf X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) then its [continuous dual space](https://bliptext.com/articles/continuous-dual-space) will be denoted by X^{\\prime}, where always holds.\nDenote the [weak-* topology](https://bliptext.com/articles/weak-topology) on X^{\\#} by and denote the [weak-* topology](https://bliptext.com/articles/weak-topology) on X^{\\prime} by The [weak-* topology](https://bliptext.com/articles/weak-topology) is also called the [topology of pointwise convergence](https://bliptext.com/articles/topology-of-pointwise-convergence) because given a map f and a [net](https://bliptext.com/articles/net-mathematics) of maps the [net](https://bliptext.com/articles/net-mathematics) f_{\\bull} converges to f in this topology if and only if for every point x in the domain, the [net](https://bliptext.com/articles/net-mathematics) of values converges to the value f(x).\n\n## Proof involving duality theory\nIf X is a [normed vector space](https://bliptext.com/articles/normed-vector-space), then the polar of a neighborhood is closed and norm-bounded in the dual space.\nIn particular, if U is the open (or closed) unit ball in X then the polar of U is the closed unit ball in the continuous dual space X^{\\prime} of X (with the [usual dual norm](https://bliptext.com/articles/operator-norm)).\nConsequently, this theorem can be specialized to:\nWhen the continuous dual space X^{\\prime} of X is an infinite dimensional normed space then it is for the closed unit ball in X^{\\prime} to be a compact subset when X^{\\prime} has its usual norm topology.\nThis is because the unit ball in the norm topology is compact if and only if the space is finite-dimensional (cf. [F. Riesz theorem](https://bliptext.com/articles/f-riesz-theorem)).\nThis theorem is one example of the utility of having different topologies on the same vector space.\nIt should be cautioned that despite appearances, the Banach–Alaoglu theorem does imply that the weak-* topology is [locally compact](https://bliptext.com/articles/locally-compact).\nThis is because the closed unit ball is only a neighborhood of the origin in the [strong topology](https://bliptext.com/articles/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.\nIn fact, it is a result of [Weil](https://bliptext.com/articles/andr-weil) that all [locally compact](https://bliptext.com/articles/locally-compact) [Hausdorff](https://bliptext.com/articles/hausdorff-space) topological vector spaces must be finite-dimensional.\n\n## Elementary proof\nThe following [elementary proof](https://bliptext.com/articles/elementary-proof) does not utilize [duality theory](https://bliptext.com/articles/duality-theory) and requires only basic concepts from set theory, topology, and functional analysis.\nWhat is needed from topology is a working knowledge of [net](https://bliptext.com/articles/net-mathematics) [convergence](https://bliptext.com/articles/net-convergence) in [topological spaces](https://bliptext.com/articles/topological-space) and familiarity with the fact that a [linear functional is continuous](https://bliptext.com/articles/continuous-linear-functional) if and only if it is [bounded on a neighborhood](https://bliptext.com/articles/continuous-linear-operator) of the origin (see the articles on [continuous linear functionals](https://bliptext.com/articles/continuous-linear-functional) and [sublinear functionals](https://bliptext.com/articles/sublinear-functional) for details).\nAlso required is a proper understanding of the technical details of how the space of all functions of the form is identified as the [Cartesian product](https://bliptext.com/articles/cartesian-product) and the relationship between [pointwise convergence](https://bliptext.com/articles/pointwise-convergence), the [product topology](https://bliptext.com/articles/product-topology), and [subspace topologies](https://bliptext.com/articles/subspace-topology) they induce on subsets such as the [algebraic dual space](https://bliptext.com/articles/algebraic-dual-space) X^{\\#} and products of subspaces such as An explanation of these details is now given for readers who are interested.\nFor every real r, will denote the closed ball of radius r centered at 0 and for any\n**Identification of functions with tuples**\nThe Cartesian product is usually thought of as the set of all X-indexed [tuples](https://bliptext.com/articles/tuple) but, since [tuples](https://bliptext.com/articles/tuple) are technically just functions from an indexing set, it can also be identified with the space of all functions having prototype as is now described:\nThis is the reason why many authors write, often without comment, the equality\nand why the Cartesian product is sometimes taken as the definition of the set of maps (or conversely).\nHowever, the Cartesian product, being the [(categorical) product](https://bliptext.com/articles/product-[category](https://bliptext.com/articles/category-mathematics)-theory) in the [category](https://bliptext.com/articles/category-mathematics) of [sets](https://bliptext.com/articles/category-of-sets) (which is a type of [inverse limit](https://bliptext.com/articles/inverse-limit)), also comes equipped with associated maps that are known as its (coordinate).\nThe at a given point z \\in X is the function\nwhere under the above identification, \\Pr{}_z sends a function to Stated in words, for a point z and function s, \"plugging z into s\" is the same as \"plugging s into \\Pr{}_z\".\nIn particular, suppose that are non-negative real numbers.\nThen where under the above identification of tuples with functions, is the set of all functions such that for every x \\in X.\nIf a subset [partitions](https://bliptext.com/articles/partition-of-a-set) X into then the linear bijection\ncanonically identifies these two Cartesian products; moreover, this map is a [homeomorphism](https://bliptext.com/articles/homeomorphism) when these products are endowed with their product topologies.\nIn terms of function spaces, this bijection could be expressed as\n**Notation for nets and function composition with nets**\nA [net](https://bliptext.com/articles/net-mathematics) in X is by definition a function from a non-empty [directed set](https://bliptext.com/articles/directed-set) (I, \\leq).\nEvery [sequence](https://bliptext.com/articles/sequence) in X, which by definition is just a function of the form \\N \\to X, is also a net.\nAs with sequences, the value of a net x_{\\bull} at an index i \\in I is denoted by x_i; however, for this proof, this value x_i may also be denoted by the usual function parentheses notation\nSimilarly for [function composition](https://bliptext.com/articles/function-composition), if F : X \\to Y is any function then the net (or sequence) that results from \"plugging x_{\\bull} into F\" is just the function although this is typically denoted by (or by if x_{\\bull} is a sequence).\nIn the proofs below, this resulting net may be denoted by any of the following notations\ndepending on whichever notation is cleanest or most clearly communicates the intended information.\nIn particular, if F : X \\to Y is continuous and in X, then the conclusion commonly written as may instead be written as or\n**Topology**\nThe set is assumed to be endowed with the [product topology](https://bliptext.com/articles/product-topology).\nIt is well known that the product topology is identical to the [topology of pointwise convergence](https://bliptext.com/articles/topology-of-pointwise-convergence).\nThis is because given f and a [net](https://bliptext.com/articles/net-mathematics) where f and every f_i is an element of then the [net](https://bliptext.com/articles/net-mathematics) [converges](https://bliptext.com/articles/convergent-net) in the product topology if and only if\nwhere because and\nthis happens if and only if\nThus converges to f in the product topology if and only if it converges to f pointwise on X.\nThis proof will also use the fact that the topology of pointwise convergence is preserved when passing to [topological subspaces](https://bliptext.com/articles/topological-subspace).\nThis means, for example, that if for every x \\in X, is some [(topological) subspace](https://bliptext.com/articles/topological-subspace) of \\mathbb{K} then the topology of pointwise convergence (or equivalently, the product topology) on is equal to the [subspace topology](https://bliptext.com/articles/subspace-topology) that the set inherits from\nAnd if S_x is closed in \\mathbb{K} for every x \\in X, then is a closed subset of\n**Characterization of **\nAn important fact used by the proof is that for any real r,\nwhere \\,\\sup\\, denotes the [supremum](https://bliptext.com/articles/supremum) and\nAs a side note, this characterization does not hold if the closed ball B_r is replaced with the open ball (and replacing with the strict inequality will not change this; for counter-examples, consider and the [identity map](https://bliptext.com/articles/identity-map) on X).\nThe essence of the Banach–Alaoglu theorem can be found in the next proposition, from which the Banach–Alaoglu theorem follows.\nUnlike the Banach–Alaoglu theorem, this proposition does require the vector space X to endowed with any topology.\nBefore proving the proposition above, it is first shown how the Banach–Alaoglu theorem follows from it (unlike the proposition, Banach–Alaoglu assumes that X is a [topological vector space](https://bliptext.com/articles/topological-vector-space) (TVS) and that U is a neighborhood of the origin).\nThe conclusion that the set is closed can also be reached by applying the following more general result, this time proved using nets, to the special case and B := B_1.\nLet and suppose that is a net in X^{\\#} the converges to f in To conclude that it must be shown that f is a linear functional.\nSo let s be a scalar and let x, y \\in X.\nFor any z \\in X, let denote\nBecause in which has the topology of pointwise convergence, in \\mathbb{K} for every z \\in X. By using in place of z, it follows that each of the following nets of scalars converges in \\mathbb{K}:\nProof that\nLet be the \"multiplication by s\" map defined by Because M is continuous and in \\mathbb{K}, it follows that where the right hand side is and the left hand side is\nwhich proves that Because also and limits in \\mathbb{K} are unique, it follows that as desired.\nProof that\nDefine a net by letting for every i \\in I. Because and it follows that in Let be the addition map defined by The continuity of A implies that in \\mathbb{K} where the right hand side is and the left hand side is\nwhich proves that Because also it follows that as desired.\n[The lemma above](https://bliptext.com/articles/) actually also follows from its corollary below since is a Hausdorff [complete uniform space](https://bliptext.com/articles/complete-uniform-space) and any subset of such a space (in particular X^{\\#}) is closed if and only if it is complete.\nBecause the underlying field \\mathbb{K} is a complete Hausdorff locally convex topological vector space, the same is true of the [product space](https://bliptext.com/articles/product-space)\nA closed subset of a complete space is complete, so by the lemma, the space is complete.\nThe above elementary proof of the Banach–Alaoglu theorem actually shows that if is any subset that satisfies (such as any [absorbing subset](https://bliptext.com/articles/absorbing-set) of X), then is a [weak-* compact](https://bliptext.com/articles/weak-topology) subset of X^{\\#}.\nAs a side note, with the help of the above elementary proof, it may be shown (see this footnote)\nthat there exist X-indexed non-negative real numbers such that\nwhere these real numbers m_{\\bull} can also be chosen to be \"minimal\" in the following sense:\nusing (so P = U^{\\#} as in the proof) and defining the notation for any if then and for every x \\in X, which shows that these numbers m_{\\bull} are unique; indeed, this [infimum](https://bliptext.com/articles/infimum) formula can be used to define them.\nIn fact, if denotes the set of all such products of closed balls containing the polar set P, then\nwhere denotes the intersection of all sets belonging to\nThis implies (among other things )\nthat the unique [least element](https://bliptext.com/articles/least-element) of with respect to this may be used as an alternative definition of this (necessarily [convex](https://bliptext.com/articles/convex-set) and [balanced](https://bliptext.com/articles/balanced-set)) set.\nThe function is a [seminorm](https://bliptext.com/articles/seminorm) and it is unchanged if U is replaced by the [convex balanced hull](https://bliptext.com/articles/convex-balanced-hull) of U (because ).\nSimilarly, because m_{\\bull} is also unchanged if U is replaced by its [closure](https://bliptext.com/articles/closure-topology) in X.\n\n# Sequential Banach–Alaoglu theorem\nA 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](https://bliptext.com/articles/separable-metric-space) normed vector space is [sequentially compact](https://bliptext.com/articles/sequentially-compact) in the weak-* topology.\nIn fact, the weak* topology on the closed unit ball of the dual of a separable space is [metrizable](https://bliptext.com/articles/metrizable), and thus compactness and sequential compactness are equivalent.\nSpecifically, let X be a separable normed space and B the closed unit ball in X^{\\prime}.\nSince X is separable, let be a countable dense subset.\nThen the following defines a metric, where for any x, y \\in B\nin which denotes the duality pairing of X^{\\prime} with X. Sequential compactness of B in this metric can be shown by a [diagonalization argument](https://bliptext.com/articles/diagonalization-argument) similar to the one employed in the proof of the [Arzelà–Ascoli theorem](https://bliptext.com/articles/arzel-ascoli-theorem).\nDue 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](https://bliptext.com/articles/partial-differential-equations) to construct solutions to PDE or [variational problems](https://bliptext.com/articles/calculus-of-variations).\nFor instance, if one wants to minimize a functional on the dual of a separable normed vector space X, one common strategy is to first construct a minimizing sequence which approaches the infimum of F, use the sequential Banach–Alaoglu theorem to extract a subsequence that converges in the weak* topology to a limit x, and then establish that x is a minimizer of F. The last step often requires F to obey a (sequential) [lower semi-continuity](https://bliptext.com/articles/semi-continuity) property in the weak* topology.\nWhen X^{\\prime} is the space of finite Radon measures on the real line (so that X = C_0(\\R) is the space of continuous functions vanishing at infinity, by the [Riesz representation theorem](https://bliptext.com/articles/riesz-markov-kakutani-representation-theorem)), the sequential Banach–Alaoglu theorem is equivalent to the [Helly selection theorem](https://bliptext.com/articles/helly-selection-theorem).\n\n# Consequences\n\n## Consequences for normed spaces\nAssume that X is a [normed space](https://bliptext.com/articles/normed-space) and endow its continuous dual space X^{\\prime} with the usual [dual norm](https://bliptext.com/articles/dual-norm).\n \n The closed unit ball in X^{\\prime} is weak-* compact.\nSo if X^{\\prime} is infinite dimensional then its closed unit ball is necessarily compact in the norm topology by [F. Riesz's theorem](https://bliptext.com/articles/f-riesz-s-theorem) (despite it being weak-* compact).\n \n A [Banach space](https://bliptext.com/articles/banach-space) is [reflexive](https://bliptext.com/articles/reflexive-space) if and only if its closed unit ball is -compact; this is known as [James' theorem](https://bliptext.com/articles/james-theorem). \n If X is a [reflexive Banach space](https://bliptext.com/articles/reflexive-banach-space), then every bounded sequence in X has a weakly convergent subsequence.\n(This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of X; or, more succinctly, by applying the [Eberlein–Šmulian theorem](https://bliptext.com/articles/eberlein-mulian-theorem).)\nFor example, suppose that X is the space [Lp space](https://bliptext.com/articles/lp-space) L^p(\\mu) where and let q satisfy\nLet be a bounded sequence of functions in X. Then there exists a subsequence and an f \\in X such that\nThe corresponding result for p = 1 is not true, as L^1(\\mu) is not reflexive. \n \n\n## Consequences for Hilbert spaces\n \n 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](https://bliptext.com/articles/reflexive-space)). \n As norm-closed, convex sets are weakly closed ([Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem)), norm-closures of convex bounded sets in Hilbert spaces or reflexive Banach spaces are weakly compact. \n Closed and bounded sets in B(H) are precompact with respect to the [weak operator topology](https://bliptext.com/articles/weak-operator-topology) (the [weak operator topology](https://bliptext.com/articles/weak-operator-topology) is weaker than the [ultraweak topology](https://bliptext.com/articles/ultraweak-topology) which is in turn the weak-* topology with respect to the predual of B(H), the [trace class](https://bliptext.com/articles/trace-class) operators).\nHence bounded sequences of operators have a weak accumulation point.\nAs a consequence, B(H) has the [Heine–Borel property](https://bliptext.com/articles/heine-borel-theorem), if equipped with either the weak operator or the ultraweak topology. \n \n\n# Relation to the axiom of choice and other statements\nThe Banach–Alaoglu may be proven by using [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem), which under the [Zermelo–Fraenkel set theory](https://bliptext.com/articles/zermelo-fraenkel-set-theory) (**ZF**) axiomatic framework is equivalent to the [axiom of choice](https://bliptext.com/articles/axiom-of-choice).\nMost mainstream functional analysis relies on **ZF** + the axiom of choice, which is often denoted by **ZF**C.\nHowever, the theorem does rely upon the axiom of choice in the [separable](https://bliptext.com/articles/separable-space) case (see [above](https://bliptext.com/articles/)): in this case there actually exists a constructive proof.\nIn the general case of an arbitrary normed space, the [ultrafilter Lemma](https://bliptext.com/articles/ultrafilter-lemma), which is strictly weaker than the axiom of choice and equivalent to Tychonoff's theorem for compact spaces, suffices for the proof of the Banach–Alaoglu theorem, and is in fact equivalent to it.\nThe Banach–Alaoglu theorem is equivalent to the [ultrafilter lemma](https://bliptext.com/articles/ultrafilter-lemma), which implies the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) for [real vector spaces](https://bliptext.com/articles/real-vector-space) (****HB****) but is not equivalent to it (said differently, Banach–Alaoglu is also strictly stronger than ****HB****).\nHowever, the [Hahn–Banach theorem](https://bliptext.com/articles/hahn-banach-theorem) is equivalent to the following weak version of the Banach–Alaoglu theorem for normed space in which the conclusion of compactness (in the [weak-* topology](https://bliptext.com/articles/weak-topology) of the closed unit ball of the dual space) is replaced with the conclusion of (also sometimes called );\n[Compactness](https://bliptext.com/articles/compact-space) implies [convex compactness](https://bliptext.com/articles/) because a topological space is compact if and only if every [family](https://bliptext.com/articles/family-of-sets) of closed subsets having the [finite intersection property](https://bliptext.com/articles/finite-intersection-property) (FIP) has non-empty intersection.\nThe [definition of [convex](https://bliptext.com/articles/convex-set) compactness](https://bliptext.com/articles/) is similar to this characterization of [compact spaces](https://bliptext.com/articles/compact-space) in terms of the FIP, except that it only involves those closed subsets that are also [convex](https://bliptext.com/articles/convex-set) (rather than all closed subsets).\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}