Locating Subsets of B(h) Relative to Seminorms Inducing the Strong-operator Topology

Let H be a Hilbert space, and A an inhabited, bounded, convex subset of B(H). We give a constructive proof that A is weak-operator totally bounded if and only if it is located relative to a certain family of seminorms that induces the strong-operator topology on B(H). This paper is a contribution to the programme of research in constructive functional analysis and operator theory. It lies entirely within a Bishop-style constructive framework ; in other words, the logic is intuitionistic, and we use an underlying set theory, such as that presented by Aczel and Rathjen [1, 2], which avoid axioms that would imply essentially nonconstructive principles such as the law of excluded middle. 1 Although carried out by strictly constructive means, our work is not insignificant within classical-logic-based computational functional analysis: each of our results and proofs is, a fortiori, classical. But constructive proofs, by their very nature, embody algorithms, and hence estimates, 2 that can be extracted—sometimes with surprising ease—and then implemented; such program-extraction and implementation can be found in Constable [8], Hayashi [9], and Schwichtenberg [13]. For example, consider our main result, Theorem 1, which deals with an inhabited, 3 bounded, convex set A of operators on an infinite-dimensional Hilbert space H. The first half of its proof is, essentially, an algorithm for converting 1 A popular alternative foundation for constructive mathematics is Martin-Löf's type theory [12]. 2 A very different approach to the extraction of estimates (often optimal ones) is adopted by Kohlenbach: working with classical logic, he uses proof-mining to extract computational information from classical proofs; see Kohlenbach [11]. 3 To say that a set is inhabited means that we can construct an element of it. This is a constructively stronger notion that nonempty (although, confusingly, some earlier work on constructive analysis uses nonempty in the sense of inhabited).

This paper is a contribution to the programme of research in constructive functional analysis and operator theory.It lies entirely within a Bishop-style constructive framework; in other words, the logic is intuitionistic, and we use an underlying set theory, such as that presented by Aczel and Rathjen [1,2], which avoid axioms that would imply essentially nonconstructive principles such as the law of excluded middle. 1lthough carried out by strictly constructive means, our work is not insignificant within classical-logic-based computational functional analysis: each of our results and proofs is, a fortiori, classical.But constructive proofs, by their very nature, embody algorithms, and hence estimates, 2 that can be extracted-sometimes with surprising ease-and then implemented; such program-extraction and implementation can be found in Constable [8], Hayashi [9], and Schwichtenberg [13].For example, consider our main result, Theorem 1, which deals with an inhabited, 3 bounded, convex set A of operators on an infinite-dimensional Hilbert space H .The first half of its proof is, essentially, an algorithm for converting 2 Douglas Bridges -finite ε-approximations to A relative to the seminorms defining the weakoperator topology on B(H) -into a computation of distances from A relative to a certain family of seminorms that induces the strong-operator topology on B(H).
The second half is an algorithm for carrying out this conversion in reverse.Of course, the practical extraction and implementation of these algorithms would be a nontrivial business; but it could be done.
We begin by recalling some definitions from the constructive theory of locally convex spaces.A subset S of a locally convex space X, (p i ) i∈I , where the p i are the seminorms defining the topology on X , is said to be located in exists for each x ∈ X and each finitely enumerable 4 subset F of I .On the other hand, S is said to be totally bounded if for each finitely enumerable subset F of I and each ε > 0, there exists a finitely enumerable subset T of S with the property that for each x ∈ S there exists y ∈ T with i∈F p i (x − y) < ε; such a set T is then called a finitely enumerable ε-approximation to S relative to the seminorm i∈F p i .
We note these facts about total boundedness and locatedness: -The image of a totally bounded set under a uniformly continuous mapping between locally convex spaces is totally bounded ([6], Proposition 5.4.2).
-Every totally bounded subset of X is located, and every located subset of a totally bounded set is totally bounded ([6], Propositions 5.4.4 and 5.4.5).
The For each integer N 2 we denote, for example, by x the N -tuple (x 1 , . . ., x N ) of elements of H , and we define H N to be the Hilbert direct sum of N copies of H .Although one frequently describes the strong-operator topology by means of the L 1like seminorms , where x ∈ H N .We say that a subset A of B(H) is k-located if it is located relative to the family of L k -like seminorms (k = 1, 2).Note that although each of the two L k -families induces the strong-operator topology on B(H), it is not a priori the case that the metric-dependent notions of 1-locatedness and 2-locatedness coincide on a given subset A of B(H).It will be a consequence of our main result, which we now state, that these two notions of locatedness do coincide when A is inhabited, bounded, and convex.
Theorem 1 Let H be an infinite-dimensional Hilbert space, and A an inhabited, bounded, convex subset of B(H).Then A is 2-located if and only if it is weakoperator totally bounded.
In the case where H is separable, the equivalence of 1-locatedness and weak-operator total boundedness for inhabited, bounded, convex subsets of B(H) was proved by Spitters ([14], Corollary 10), who took a non-elementary route through trace-class operators and normal states.In the non-separable case, the implication from weakoperator total boundedness to 1-locatedness is proved by Bridges, Ishihara and Vît ¸ȃ [7] (Theorem 3.8), using general results about infima of real-valued continuous functions on convex sets in normed spaces (a counterpart of which plays a role in our work below).
We shall prove Theorem 1 without separability and by relatively elementary methods.Before doing so, we remind ourselves of a common construction and deal with some preliminary results.The complicated proof of the first of these, due to Ishihara, can be found in [10] (Corollary 5) or Bridges and Vît ¸ȃ [6] (Corollary 6.2.9).
Proposition 2 Let C be an inhabited, bounded, convex subset of an inner product space H . Then C is located if and only if sup {Re x, y : y ∈ C} exists for each x ∈ H .
Our second preliminary result is a version of a classically trivial result about Banach spaces ([6], Proposition 5.3.4),whose known constructive proof is not trivial as it uses the Hahn-Banach theorem.However, in the case where X is a Hilbert space, there is a natural, more elementary proof, for which we need two items of information about dimensionality in a normed space X .First, we note that every finite-dimensional subspace of X is located ( [6], Lemma 4.1.2).Secondly, we say that X is infinitedimensional if for each finite-dimensional subspace V of X , there exists x ∈ X with ρ (x, V) > 0 (in which case the orthogonal complement of V contains a unit vector).For additional material on finite-and infinite-dimensionality in normed spaces, see Chapter 4 of Bridges and Vît ¸ȃ [6].
Lemma 3 Let H be an infinite-dimensional Hilbert space, and let x 1 , . . ., x N be vectors in H . Then for each t > 0, there exist pairwise orthogonal unit vectors e 1 , . . ., e N in H such that the vectors x n ≡ x n + te n (1 n N) are linearly independent.
Proof To begin with, pick a unit vector e 1 such that x 1 ≡ x 1 + te 1 = 0. Suppose that for some n < N we have found the desired vectors e 1 , . . ., e n , and let V be the n-dimensional subspace of H generated by the vectors Either ρ x n+1 , V > 0 or ρ x n+1 , V < t.In the first case, V ∪ {x n+1 } generates an (n + 1)-dimensional subspace W of H , and we can pick a unit vector e orthogonal to W . Then for each v ∈ V, tρ(e, W) = t.
Hence ρ x n+1 + te, V t > 0, so x n+1 + te is linearly independent of V , and we can take e n+1 ≡ e.
In the case where ρ x n+1 , V < t, we pick a unit vector e orthogonal to V .With P the projection of H on V , and I the identity operator on H , we have Hence ρ x n+1 + te, V > 0, so x n+1 + te is linearly independent of V , and we can take e n+1 ≡ e.

Returning to the set-up of Theorem 1, for each T ∈ B(H) define
Tx ≡ (Tx 1 , . . ., Tx N ) , and for any subset A of B(H) define Lemma 4 If A is an inhabited, bounded, 2-located subset of B(H), and x ∈ H N , then Proof We may assume that A ⊂ B 1 (H).Let 0 < α < β , and set ε ≡ 1 3 (β − α).By Lemma 3, since H is infinite-dimensional, there exist pairwise orthogonal unit vectors e 1 , . . ., e N such that the vectors are linearly independent.Given y ∈ H N , construct S ∈ B(H) such that Sx n = y n for each n.(This is possible since the locatedness of the n-dimensional span V of {x 1 , . . ., x N } implies the existence of the projection P of H onto V , and hence enables us to set exists.Either λ > α + ε or λ < β − ε.In the former case, for each T ∈ A we have In the case λ < β − ε, there exists T ∈ A such that It now follows from the constructive greatest-lower-bound principle ( The following lemma is similar to Lemma 3.2 of Bridges and Vît ¸ȃ [7], and is needed to remove a preliminary restriction in part of the proof of Theorem 1.
Lemma 5 Let f 1 , . . ., f N be bounded, nonnegative functions on a set S such that for each δ > 0, Since m δ exists, we can find x 0 ∈ S such that Since ε > 0 is arbitrary, it follows that inf N n=1 f n (x) 2 : x ∈ X exists; whence the desired infimum also exists.
We now give the proof of Theorem 1.
Proof Assume that A is 2-located in B(H).Let N be any positive integer, and define H N , T, A as above.Then A is an inhabited, bounded, convex subset of B(H N ).By Lemma 4, for each x ∈ H N the inhabited, bounded, convex set Again applying Proposition 2, we see that S x,y is located in C N , regarded as a Hilbert space over C; being also bounded, S x,y is therefore totally bounded.Since all norms on C N are equivalent, it follows that for each ε > 0, there exists a finitely enumerable subset {T 1 , . . ., T m } of A such that the elements form a finitely enumerable ε-approximation to S x,y relative to the norm Then f is a convex function.In view of (2) and Lemma 3.6 of Bridges, Ishihara and Vît ¸ȃ [7], we see that the mappings S − T x (S − T) x n are uniformly differentiable on C, and hence (again note (2)) that f is also.It follows from Theorem 2.2 of the same reference that the infimum in (1) exists.
We now remove the condition (2).Let H denote the direct sum H ⊕ H of two copies of H , let δ > 0, and let A ≡ A ⊕ δ 1/2 I , where I is the identity operator on H and T ⊕ δ 1/2 I (x, y) ≡ Tx, δ 1/2 y (T ∈ B(H); x, y ∈ H) .Referring to Spitters [14] (Corollary 10) and Bridges, Ishihara and Vît ¸ȃ [7] (Theorem 8), we immediately obtain Corollary 6 Let H be an infinite-dimensional Hilbert space, and A an inhabited, bounded, convex subset of B(H).Then A is 1-located if and only if it is 2-located.
where x ∈ H N , in this paper we focus our attention on an alternative family of seminorms inducing τ s : namely, the family of L 2 -like seminorms

Ax≡
Tx : T ∈ A is located in H N .It follows from Proposition 2 that for all x, y in H N , σ x,y ≡ sup Re Tx, y : T ∈ A = sup Re y, Tx : T ∈ A exists.Now, S x,y ≡ Tx 1 , y 1 , . . ., Tx N , y N : T ∈ A Douglas Bridges is an inhabited, bounded, and convex subset of the Hilbert space C N , taken with the usual inner product.Moreover, for each η ∈ C N , sup {Re η, ζ : ζ ∈ S x,y } = sup Re N k=1 η k ζ k : ζ ∈ S x,y = sup Re N k=1 η k y k , Tx k : T ∈ A = σ x,z exists, where z ≡ (η 1 y 1 , . . ., η N y N ) ∈ H N .

(ζ 1
, . . ., ζ N ) N n=1 |ζ n | on C N .Hence for each T ∈ A there exists k m such that N n=1 | (T − T k ) x n , y n | < ε.Thus {T k : 1 k m} is a finitely enumerable ε-approximation to A relative to the seminorm T N n=1 | Tx k , y k |.It follows that A is weak-operator totally bounded.To prove the converse, assume that A is weak-operator totally bounded.Let S ∈ B(H) and x ∈ H N .We need to prove that

Define 2 =
S ∈ B(H ) by S (x, y) ≡ (Sx, 0).Fix a unit vector e ∈ H , and let x n ≡ (x n , e) (1 n N).Then for each n N and each T ∈ A,S x n − T, δ 1/2 x n Sx n − Tx n 2 + δ δ, so ρ S x n , A x n > 0.It is easy to verify that A is weak-operator totally bounded.Applying the first part of the proof to A , S , and x , we see that m δ ≡ inf δ > 0 is arbitrary, it follows from Lemma 5 that the infimum at (1) exists in the general case.Since S and x are arbitrary, we conclude that A is 2-located.