Banach–Alaoglu theorem explained

In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology.^[1] A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

This 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.

History

According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe most important fact about the weak-* topology—[that] echos throughout functional analysis.” In 1912, Helly proved that the unit ball of the continuous dual space of

C([a, b]) is countably weak-* compact. In 1932, Stefan Banach proved that the closed unit ball in the continuous dual space of any separable normed space is sequentially weak-* compact (Banach only considered sequential compactness). The proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu. According to Pietsch [2007], there are at least twelve mathematicians who can lay claim to this theorem or an important predecessor to it.

The Bourbaki–Alaoglu theorem is a generalization^{[2] [3]} of the original theorem by Bourbaki to dual topologies on locally convex spaces. This theorem is also called the Banach–Alaoglu theorem or the weak-* compactness theorem and it is commonly called simply the Alaoglu theorem.

Statement

See also: Dual system and Polar set.

If

X is a vector space over the field K then X ^{\#} will denote the algebraic dual space of X and these two spaces are henceforth associated with the bilinear \left\langle ⋅ , ⋅ \right\rangle : X x X ^{\#} \to K defined by $$\backslash left\backslash langle\; x,\; f\; \backslash right\backslash rangle\; ~\backslash stackrel~\; f(x)$$where the triple \left\langle X, X ^{\#} ,\left\langle ⋅ , ⋅ \right\rangle \right\rangle forms a dual system called the .

If

X is a topological vector space (TVS) then its continuous dual space will be denoted by X ^{\prime} , where X ^{\prime} \subseteq X ^{\#} always holds. Denote the weak-* topology on X ^{\#} by \sigma\left(X ^{\#} , X\right) and denote the weak-* topology on X ^{\prime} by \sigma\left(X ^{\prime} , X\right). The weak-* topology is also called the topology of pointwise convergence because given a map f and a net of maps f _{\bull} = \left(f _{i\right)} _{i} , the net f _{\bull} converges to f in this topology if and only if for every point x in the domain, the net of values \left(f _{i(x)\right)} _{i} converges to the value f(x).

Proof involving duality theory

If

X is a normed vector space, then the polar of a neighborhood is closed and norm-bounded in the dual space. In 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). Consequently, this theorem can be specialized to:

When 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. 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 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 elementary proof does not utilize duality theory and requires only basic concepts from set theory, topology, and functional analysis. What is needed from topology is a working knowledge of net convergence in topological spaces and familiarity with the fact that a linear functional is continuous if and only if it is bounded on a neighborhood of the origin (see the articles on continuous linear functionals and sublinear functionals for details). Also required is a proper understanding of the technical details of how the space

K ^{X} of all functions of the form X \to K is identified as the Cartesian product $\backslash prod\_\; \backslash mathbb,$ and the relationship between pointwise convergence, the product topology, and subspace topologies they induce on subsets such as the algebraic dual space X ^{\#} and products of subspaces such as $\backslash prod\_\; B\_.$ An explanation of these details is now given for readers who are interested.

For every real

r, B _{r} ~\stackrel{\scriptscriptstyle def }~ \ will denote the closed ball of radius r centered at 0 and r U ~\stackrel{\scriptscriptstyle def }~ \ for any U \subseteq X,

Identification of functions with tuples

The Cartesian product $\backslash prod\_\; \backslash mathbb$ is usually thought of as the set of all

X -indexed tuples s _{\bull} = \left(s _{x\right)} _{x} but, since tuples are technically just functions from an indexing set, it can also be identified with the space K ^{X} of all functions having prototype X \to K , as is now described:

• : A function

s : X \to K belonging to K ^{X} is identified with its (X -indexed) "" s _{\bull} ~\stackrel{\scriptscriptstyle def }~ (s(x))_.

• : A tuple

s _{\bull} = \left(s _{x\right)} _{x} in $\backslash prod\_\; \backslash mathbb$ is identified with the function s : X \to K defined by s(x) ~\stackrel{\scriptscriptstyle def }~ s_x; this function's "tuple of values" is the original tuple \left(s _{x\right)} _{x} .

This is the reason why many authors write, often without comment, the equality $$\backslash mathbb^X\; =\; \backslash prod\_\; \backslash mathbb$$and why the Cartesian product $\backslash prod\_\; \backslash mathbb$ is sometimes taken as the definition of the set of maps

K ^{X} (or conversely). 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) .

The at a given point

z \in X is the function$$\backslash Pr\_z\; :\; \backslash prod\_\; \backslash mathbb\; \backslash to\; \backslash mathbb\; \backslash quad\; \backslash text\; \backslash quad\; s\_\; =\; \backslash left(s\_x\backslash right)\_\; \backslash mapsto\; s\_z$$where under the above identification, \Pr{} _{z} sends a function s : X \to K to$$\backslash Pr\_z(s)\; ~\backslash stackrel~\; s(z).$$ Stated in words, for a point z and function s, "plugging z into s " is the same as "plugging s into \Pr{} _{z} ".

In particular, suppose that

\left(r _{x\right)} _{x} are non-negative real numbers. Then \prod _{x} B _{r _{x}} \subseteq \prod _{x} K = K ^{X,} where under the above identification of tuples with functions, \prod _{x} B _{r _{x}} is the set of all functions s \in K ^{X} such that s(x) \in B _{r _{x}} for every x \in X.

If a subset

U \subseteq X partitions X into X = U \cup (X \setminus U) then the linear bijectioncanonically identifies these two Cartesian products; moreover, this map is a homeomorphism when these products are endowed with their product topologies. In terms of function spaces, this bijection could be expressed as

Notation for nets and function composition with nets

x _{\bull} = \left(x _{i\right)} _{i} in X is by definition a function x _{\bull} : I \to X from a non-empty directed set (I, \leq). Every sequence in X, which by definition is just a function of the form \N \to X, is also a net. As 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 x _{\bull} (i). Similarly for 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 F \circ x _{\bull} : I \to Y, although this is typically denoted by \left(F\left(x _{i\right)\right)} _{i} (or by \left(F\left(x _{i\right)\right)} ^{inf ty} ^{i =1} if x _{\bull} is a sequence). In the proofs below, this resulting net may be denoted by any of the following notations $$F\backslash left(x\_\backslash right)\; =\; \backslash left(F\backslash left(x\_i\backslash right)\backslash right)\_\; ~\backslash stackrel~\; F\; \backslash circ\; x\_,$$ depending on whichever notation is cleanest or most clearly communicates the intended information. In particular, if F : X \to Y is continuous and x _{\bull} \to x in X, then the conclusion commonly written as \left(F\left(x _{i\right)\right)} _{i} \to F(x) may instead be written as F\left(x _{\bull} \right) \to F(x) or F \circ x _{\bull} \to F(x).

Topology

The set $\backslash mathbb^X\; =\; \backslash prod\_\; \backslash mathbb$ 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

f and a net \left(f _{i\right)} _{i} , where f and every f _{i} is an element of $\backslash mathbb^X\; =\; \backslash prod\_\; \backslash mathbb,$ then the net \left(f _{i\right)} _{i} \to f converges in the product topology if and only if

for every

z \in X, the net \Pr{} _{z\left(\left(f} _{i\right)} _{i} \right) \to \Pr{} _{z(f)} converges in K ,

where because

\Pr{} _{z(f)} = f(z) and $\backslash Pr\_z\backslash left(\backslash left(f\_i\backslash right)\_\backslash right)\; ~\backslash stackrel~\; \backslash left(\backslash Pr\_z\backslash left(f\_i\backslash right)\backslash right)\_\; =\; \backslash left(f\_i(z)\backslash right)\_,$ this happens if and only if

for every

z \in X, the net \left(f _{i(z)\right)} _{i} \to f(z) converges in K ,

Thus

\left(f _{i\right)} _{i} converges to f in the product topology if and only if it converges to f pointwise on X.

This proof will also use the fact that the topology of pointwise convergence is preserved when passing to topological subspaces. This means, for example, that if for every

x \in X, S _{x} \subseteq K is some (topological) subspace of K then the topology of pointwise convergence (or equivalently, the product topology) on $\backslash prod\_\; S\_x$ is equal to the subspace topology that the set $\backslash prod\_\; S\_x$ inherits from $\backslash prod\_\; \backslash mathbb.$ And if S _{x} is closed in K for every x \in X, then $\backslash prod\_\; S\_x$ is a closed subset of $\backslash prod\_\; \backslash mathbb.$

Characterization of

\sup _{u} |f(u)| \leq r

An important fact used by the proof is that for any real

r, $$\backslash sup\_\; |f(u)|\; \backslash leq\; r\; \backslash qquad\; \backslash text\; \backslash qquad\; f(U)\; \backslash subseteq\; B\_r$$where \sup denotes the supremum and f(U) ~\stackrel{\scriptscriptstyle def }~ \. As a side note, this characterization does not hold if the closed ball B _{r} is replaced with the open ball \{c \in K : |c| < r\} (and replacing \sup _{u} |f(u)| \leq r with the strict inequality \sup _{u} |f(u)| < r will not change this; for counter-examples, consider X ~\stackrel{\scriptscriptstyle def }~ \mathbb and the identity map f ~\stackrel{\scriptscriptstyle def }~ \operatorname on X ).

The essence of the Banach–Alaoglu theorem can be found in the next proposition, from which the Banach–Alaoglu theorem follows. Unlike the Banach–Alaoglu theorem, this proposition does require the vector space

X to endowed with any topology.

Before 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 (TVS) and that U is a neighborhood of the origin).

The conclusion that the set

U _{B _{1}} = \left\{f \in K ^{X} : f(U) \subseteq B _{1\right\}} is closed can also be reached by applying the following more general result, this time proved using nets, to the special case Y : = K and B : = B _{1.}

Observation: If

U \subseteq X is any set and if B \subseteq Y is a closed subset of a topological space Y, then U _{B} ~\stackrel{\scriptscriptstyle def }~ \left\ is a closed subset of Y ^{X} in the topology of pointwise convergence.

Proof of observation: Let

f \in Y ^{X} and suppose that \left(f _{i\right)} _{i} is a net in U _{B} that converges pointwise to f. It remains to show that f \in U _{B,} which by definition means f(U) \subseteq B. For any u \in U, because \left(f _{i(u)\right)} _{i} \to f(u) in Y and every value f _{i(u)} \in f _{i(U)} \subseteq B belongs to the closed (in Y ) subset B, so too must this net's limit belong to this closed set; thus f(u) \in B, which completes the proof. \blacksquare

Let

f \in K ^{X} and suppose that f _{\bull} = \left(f _{i\right)} _{i} is a net in X ^{\#} the converges to f in K ^{X.} To conclude that f \in X ^{\#} , it must be shown that f is a linear functional. So let s be a scalar and let x, y \in X.

For any

z \in X, let f _{\bull} (z) : I \to K denote $$f\_(z)\; ~\backslash stackrel~\; \backslash left(f\_i(z)\backslash right)\_.$$Because f _{\bull} \to f in K ^{X,} which has the topology of pointwise convergence, f _{\bull} (z) \to f(z) in K for every z \in X. By using x, y, sx, and x + y, in place of z, it follows that each of the following nets of scalars converges in K : $$f\_(x)\; \backslash to\; f(x),\; \backslash quad\; f\_(y)\; \backslash to\; f(y),\; \backslash quad\; f\_(x\; +\; y)\; \backslash to\; f(x\; +\; y),\; \backslash quad\; \backslash text\; \backslash quad\; f\_(sx)\; \backslash to\; f(sx).$$

Proof that

f(s x) = s f(x): Let M : K \to K be the "multiplication by s " map defined by M(c) ~\stackrel{\scriptscriptstyle def }~ s c. Because M is continuous and f _{\bull} (x) \to f(x) in K , it follows that M\left(f _{\bull} (x)\right) \to M(f(x)) where the right hand side is M(f(x)) = s f(x) and the left hand side iswhich proves that f _{\bull} (sx) \to s f(x). Because also f _{\bull} (sx) \to f(sx) and limits in K are unique, it follows that s f(x) = f(s x), as desired.

Proof that

f(x + y) = f(x) + f(y): Define a net z _{\bull} = \left(z _{i\right)} _{i} : I \to K x K by letting z _{i} ~\stackrel{\scriptscriptstyle def }~ \left(f_i(x), f_i(y)\right) for every i \in I. Because f _{\bull} (x) = \left(f _{i(x)\right)} _{i} \to f(x) and f _{\bull} (y) = \left(f _{i(y)\right)} _{i} \to f(y), it follows that z _{\bull} \to ( f(x), f(y) ) in K x K . Let A : K x K \to K be the addition map defined by A(x, y) ~\stackrel{\scriptscriptstyle def }~ x + y. The continuity of A implies that A\left(z _{\bull} \right) \to A(f(x), f(y)) in K where the right hand side is A(f(x), f(y)) = f(x) + f(y) and the left hand side is$$A\backslash left(z\_\backslash right)\; ~\backslash stackrel~\; A\; \backslash circ\; z\_\; =\; \backslash left(A\backslash left(z\_i\backslash right)\backslash right)\_\; =\; \backslash left(A\backslash left(f\_i(x),\; f\_i(y)\backslash right)\backslash right)\_\; =\; \backslash left(f\_i(x)\; +\; f\_i(y)\backslash right)\_\; =\; \backslash left(f\_i(x\; +\; y)\backslash right)\_\; =\; f\_(x\; +\; y)$$which proves that f _{\bull} (x + y) \to f(x) + f(y). Because also f _{\bull} (x + y) \to f(x + y), it follows that f(x + y) = f(x) + f(y), as desired. \blacksquare

The lemma above actually also follows from its corollary below since

\prod _{x} K is a Hausdorff complete uniform space and any subset of such a space (in particular X ^{\#} ) is closed if and only if it is complete.

Because the underlying field

K is a complete Hausdorff locally convex topological vector space, the same is true of the product space $\backslash mathbb^X\; =\; \backslash prod\_\; \backslash mathbb.$ A closed subset of a complete space is complete, so by the lemma, the space \left(X ^{\#} , \sigma\left(X ^{\#} , X\right)\right) is complete.\blacksquare

The above elementary proof of the Banach–Alaoglu theorem actually shows that if

U \subseteq X is any subset that satisfies X = ( 0 , inf ty) U ~\stackrel{\scriptscriptstyle def }~ \ (such as any absorbing subset of X ), then U ^{\#} ~\stackrel{\scriptscriptstyle def }~ \left\ is a weak-* compact subset of X ^{\#} .

As a side note, with the help of the above elementary proof, it may be shown (see this footnote)^[4]

Notes and References

1. , Theorem 3.15.
2. , Theorem (4) in §20.9.
3. , Theorem 23.5.
4. Bell. J.. Fremlin. David. A Geometric Form of the Axiom of Choice. Fundamenta Mathematicae. 1972. 77. 2. 167–170. 10.4064/fm-77-2-167-170. 26 Dec 2021.
5. This tuple m _{\bull} ~\stackrel{\scriptscriptstyle def }~ \left(m_x\right)_ is the least element of T _{P} with respect to natural induced pointwise partial order defined by R _{\bull} \leq S _{\bull} if and only if R _{x} \leq S _{x} for every x \in X. Thus, every neighborhood U of the origin in X can be associated with this unique (minimum) function m _{\bull} : X \to [ 0 , inf ty). For any x \in X, if r > 0 is such that x \in r U then m _{x} \leq r so that in particular, m _{0} = 0 and m _{u} \leq 1 for every u \in U. ) that $\backslash prod\; B\_\; =\; \backslash prod\_\; B\_$ the unique least element of \operatorname{Box} _{P} with respect to \subseteq; this may be used as an alternative definition of this (necessarily convex and balanced) set. The function m _{\bull} ~\stackrel{\scriptscriptstyle def }~ \left(m_x\right)_ : X \to [0, \infty)</math> is a [[seminorm]] and it is unchanged if U is replaced by the convex balanced hull of U (because U ^{\#} = [\operatorname{cobal} U] ^{\#} ). Similarly, because U ^{\circ} = \left[\operatorname{cl} _{X} U\right] ^{\circ} , m _{\bull} is also unchanged if U is replaced by its closure in X.

   Sequential Banach–Alaoglu theorem

   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

   X be a separable normed space and B the closed unit ball in X ^{\prime} . Since X is separable, let x _{\bull} = \left(x _{n\right)} ^{inf ty} ^{n =1} be a countable dense subset. Then the following defines a metric, where for any x, y \in B $$\backslash rho(x,y)\; =\; \backslash sum\_^\backslash infty\; \backslash ,\; 2^\; \backslash ,\; \backslash frac$$in which \langle ⋅ , ⋅ \rangle denotes the duality pairing of X ^{\prime} with X. Sequential compactness of B 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

   F : X ^{\prime} \to \R on the dual of a separable normed vector space X, one common strategy is to first construct a minimizing sequence x _{1,} x _{2,} \ldots \in X ^{\prime} 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 property in the weak* topology.

   When

   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), the sequential Banach–Alaoglu theorem is equivalent to the Helly selection theorem.

   Consequences

   Consequences for normed spaces

   Assume that

   X is a normed space and endow its continuous dual space X ^{\prime} with the usual dual norm.
   • The closed unit ball in
     X ^{\prime} is weak-* compact. So if X ^{\prime} is infinite dimensional then its closed unit ball is necessarily compact in the norm topology by F. Riesz's theorem (despite it being weak-* compact).
   • A Banach space is reflexive if and only if its closed unit ball is \sigma\left(X, X ^{\prime} \right) -compact; this is known as James' theorem.
   • If X is a reflexive Banach space, then every bounded sequence in X has a weakly convergent subsequence. (This follows by applying the Banach–Alaoglu theorem to a weakly metrizable subspace of X ; or, more succinctly, by applying the Eberlein–Šmulian theorem.) For example, suppose that X is the space Lp space L ^{p(\mu)} where 1 < p < inf ty and let q satisfy (1)/(p) + (1)/(q) = 1 . Let f _{1,} f _{2,} \ldots be a bounded sequence of functions in X. Then there exists a subsequence \left(f _{n _{k}} \right) ^{inf ty} _{k =1} and an f \in X such that$$\backslash int\; f\_\; g\backslash ,d\backslash mu\; \backslash to\; \backslash int\; f\; g\backslash ,d\backslash mu\; \backslash qquad\; \backslash text\; g\; \backslash in\; L^q(\backslash mu)\; =\; X^.$$The corresponding result for p = 1 is not true, as L ^{1(\mu)} is not reflexive.

   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
     B(H) 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 B(H), the trace class operators). Hence bounded sequences of operators have a weak accumulation point. As a consequence, B(H) has the Heine–Borel property, if equipped with either the weak operator or the ultraweak topology.

   Relation to the axiom of choice and other statements

   See also: Krein–Milman theorem#Relation to other statements.

   The Banach–Alaoglu may be proven by using Tychonoff's theorem, which under the Zermelo–Fraenkel set theory (ZF) axiomatic framework is equivalent to the axiom of choice. Most mainstream functional analysis relies on ZF + the axiom of choice, which is often denoted by ZFC. However, the theorem does rely upon the axiom of choice in the separable case (see above): in this case there actually exists a constructive proof. In the general case of an arbitrary normed space, the 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.

   The Banach–Alaoglu theorem is equivalent to the ultrafilter lemma, which implies the Hahn–Banach theorem for real vector spaces (HB) but is not equivalent to it (said differently, Banach–Alaoglu is also strictly stronger than HB). However, the Hahn–Banach theorem is equivalent to the following weak version of the Banach–Alaoglu theorem for normed space^[4] in which the conclusion of compactness (in the weak-* topology of the closed unit ball of the dual space) is replaced with the conclusion of (also sometimes called);

   Compactness implies convex compactness because a topological space is compact if and only if every family of closed subsets having the finite intersection property (FIP) has non-empty intersection. The definition of convex compactness is similar to this characterization of compact spaces in terms of the FIP, except that it only involves those closed subsets that are also convex (rather than all closed subsets).

   Notes

   Proofs

   References

   • • Book: Meise. Reinhold. Vogt. Dietmar. Introduction to Functional Analysis. Clarendon Press. Oxford, England. 1997. 0-19-851485-9. Theorem 23.5. 264.
     • See Theorem 3.15, p. 68.

   Further reading

6. For any non-empty subset A \subseteq [ 0 , inf ty), the equality \cap \left\{B _{a} : a \in A\right\} = B _{inf A} holds (the intersection on the left is a closed, rather than open, disk − possibly of radius 0 − because it is an intersection of closed subsets of K and so must itself be closed). For every x \in X, let m _{x} = inf \left\{ R _{x} : R _{\bull} \in T _{P} \right\} so that the previous set equality implies \cap \operatorname{Box} _{P} = c ap _{R _{\bull} \in T _{P}} \prod _{x} B _{R _{x}} = \prod _{x} c ap _{R _{\bull} \in T _{P}} B _{R _{x}} = \prod _{x} B _{m _{x}} . From P \subseteq \cap \operatorname{Box} _{P} it follows that m _{\bull} \in T _{P} and \cap \operatorname{Box} _{P} \in \operatorname{Box} _{P,} thereby making \cap \operatorname{Box} _{P} the least element of \operatorname{Box} _{P} with respect to \subseteq. (In fact, the family \operatorname{Box} _{P} is closed under (non-nullary) arbitrary intersections and also under finite unions of at least one set). The elementary proof showed that T _{P} and \operatorname{Box} _{P} are not empty and moreover, it also even showed that T _{P} has an element \left(r _{x\right)} _{x} that satisfies r _{u} = 1 for every u \in U, which implies that m _{u} \leq 1 for every u \in U. The inclusion P ~\subseteq~ \left(\cap \operatorname{Box} _{P\right)} \cap X ^{\prime} ~\subseteq~ \left(\cap \operatorname{Box} _{P\right)} \cap X ^{\#} is immediate; to prove the reverse inclusion, let f \in \left(\cap \operatorname{Box} _{P\right)} \cap X ^{\#} . By definition, f \in P ~\stackrel{\scriptscriptstyle def }~ U^ if and only if \sup _{u} |f(u)| \leq 1 , so let u \in U and it remains to show that |f(u)| \leq 1 . From f \in \cap \operatorname{Box} _{P} = \prod B _{m _{\bull}} , it follows that f(u) = \Pr{} _{u} (f) \in \Pr{} _{u} \left(\prod _{x} B _{m _{x}} \right) = B _{m _{u}} , which implies that |f(u)| \leq m _{u} \leq 1 , as desired. \blacksquare that there exist X -indexed non-negative real numbers m _{\bull} = \left(m _{x\right)} _{x} such that where these real numbers m _{\bull} can also be chosen to be "minimal" in the following sense: using P ~\stackrel{\scriptscriptstyle def }~ U^ (so P = U ^{\#} as in the proof) and defining the notation \prod B _{R _{\bull}} ~\stackrel{\scriptscriptstyle def }~ \prod_ B_ for any R _{\bull} = \left(R _{x\right)} _{x} \in \R ^{X,} if$$T\_P\; ~\backslash stackrel~\; \backslash left\backslash$$ then m _{\bull} \in T _{P} and for every x \in X, m _{x} = inf \left\{ R _{x} : R _{\bull} \in T _{P} \right\}, which shows that these numbers m _{\bull} are unique; indeed, this infimum formula can be used to define them.

   In fact, if

   \operatorname{Box} _{P} denotes the set of all such products of closed balls containing the polar set P, $$\backslash operatorname\_P\; ~\backslash stackrel~\; \backslash left\backslash \; ~=~\; \backslash left\backslash ,$$ then $\backslash prod\; B\_\; =\; \backslash cap\; \backslash operatorname\_P\; \backslash in\; \backslash operatorname\_P$ where $\backslash bigcap\; \backslash operatorname\_P$ denotes the intersection of all sets belonging to \operatorname{Box} _{P.}

   This implies (among other things^[4]