[Math.FA] 15 Dec 2014 Rigged Hilbert Spaces and Inductive Limits

Home , Hilbert space

arXiv:1412.5092v1 [math.FA] 15 Dec 2014 iuoneadSchwartz. Dieudonné and omts where commutes, hr xsssc httediagram the that such exists there iiso pcr ffiiedmninlHletsae.As efilteg the fill we [12]. Also in presented spaces. in construction strict Hilbert to the finite-dimensional ourselves of p Rec restricting the spectra by quant In [2] 6]. of of in limits [2]. [10, method RHS functions states the construct to simplify analytic resonant we proposed Hilbert of note was of rigged limits spaces inductive of theory on of Constructions use based the the are review). expa in physics a eigenfunction as in for of used well 7] theory [5, as the (see 13] in mechanics role 3, important [11, an sions play now 1955 in uhthat such ihcniuu maps continuous with { rpsto 1. Proposition proper universality analogous have [8]. we limit 2.3.5) direct [4], II.6.1; ([14], topology i ϕ embeddings ,te = Φ then ), H ij ∗ ϕ i igdHletsae n nutv limits inductive and spaces Hilbert Rigged igdHletsae RS nrdcdb efn n Kostyuchen and Gelfand by introduced (RHS) spaces Hilbert Rigged Taking eal[]ta a that [8] Recall Suppose [email protected] } jk eoe ylim by denoted tuto yBloot n Trapani. and Bellomonte by struction (Φ iesoa usae faHletspace Hilbert a of subspaces dimensional = H, , ewrs igdHletsae nutv ii,LB-space limit, inductive space, Hilbert rigged Keywords: 46A13 47A70 2010: MSC ecntutancersaeΦa nidcielmto finite- of limit inductive an as Φ space nuclear a construct We ϕ ϕ ik = Φ i i ˜ Φ { ϕ +1 od o all for holds ′ H eoe igdHletsae hssmlfigtecon- the simplifying thus space, Hilbert rigged a becomes ) i S ϕ i ˜ : } n ∞ i Let − C lim = Φ H sa naigsqec fLV (i.e. LCVS of sequence inflating an is ˜ = 1 ietspectrum direct H i → nPo.1w bantefloigcrlaydeto due corollary following the obtain we 1 Prop. in Φ ϕ ˜ → i . . . H i Φ Φ ϕ ˜ +1 eaLV n let and LCVS a be noe ihidcietplg ...canonical w.r.t. topology inductive with endowed i ij scalled is Φ `❆ hndet elkonuieslt finductive of universality well-known to due Then . h◗ o ❆ → ϕ k < j < i ◗ ˜ ❆ o all for : ◗ i ❆ ◗ ❆ H ❆ ◗ ❆ H ◗ ..Pol’shin S.A. ❆ i ◗ / . H ◗ i ◗ ϕ ˜ → ◗ Abstract i i +1 ◗ i ❧ hntecniuu map continuous the Then . ◗ ϕ ❧ . safamily a is ◗ H ❧ i i ◗ nutv limit inductive ❧ 1 λ ◗ +1 ❧ j ◗ ❧ / ❧ H o all for ❧ ϕ ❧ i i +1 ❧ ϕ ˜ ❧ ③ ❧ i ③ ❧ ③ : ∗ ❧ ③ { ❧ ③ j < i ϕ H ③ H H ❧ ③ i ❧ +1 ③ i ❧ i ③ / ; 6 < . . . nsc a that way a such in → i ∈ ftedrc spectrum direct the of uhta h equality the that such Φ ˜ N ecniuu maps continuous be } H fLV together LCVS of i ⊂ H λ i Φ : +1 ductive spaces o all for resent pin ap ently → yof ty um (1) ko n- Φ ˜ Corollary 1. Any linear functional on lim→ Hi is continuous iff its restric- tions to each Hi are.

Suppose now that Hi =6 Hi+1 for all i and the topology of Hi is equivalent to those induced from Hi+1, then the inductive limit is called strict. It is well known that the strict inductive limit is complete, Hausdorff and nuclear provided all the Hi are ([14], II.6.4, II.6.6, III.7.4; [4], 2.6.2, 2.6.5). Definition 1. Let i : Φ → H be an injective homomorphism of complete nuclear space Φ into Hilbert space H with dense range. Then (Φ,H, Φ′) is called Gelfand triple (or RHS). Example 1. Consider the construction of [12] based on the embedding of the space s of rapidly decreasing sequences into ℓ2. The range of this embedding is indeed dense in ℓ2 (each finite-dimensional subspace of ℓ2 contains an element of s). It remains to show that this embedding is continuous, but it follows from the following definition of topology of s by seminorms equivalent to ordinary one ([4], 1.3.19): ∞ 2 4k 2 qk(x) = X n |xn| . n=1 In particular, we can take functional realization s ∼= C∞(T) and ℓ2 ∼= L2(T). Let H ∼= ℓ2 be a separable infinite-dimensional Hilbert space with basis ∞ e , i N x x e H { i ∈ }, then for any = Pi=1 i i ∈ we can define its projections P x n x e H e , i ,...,n α , i N n = Pi=1 i i ∈ n = span{ i = 1 }. Let { i ∈ } be an increasing sequence of natural numbers, we see that Φ = lim→ Hαi is dense in H since kx − PnxkH →n→∞ 0. Then from Prop. 1 it follows that the continuous map λ : Φ → H there exist and it is injective. Indeed, otherwise we have λϕαi (x) = 0 for some i and x =6 0 in Hαi , but this violates the injectivity ofϕ ˜i in the diagram (1). The above construction may be reversed in the following sense. Let Φ = lim→ Hi be a strict inductive limit of a sequence of finite-dimensional

Hilbert spaces. Consider the bilinear form on Φ defined by hx, yiΦ = hx, yiHi for x, y ∈ Hi. This definition is correct since all the ϕij are isometries and h−, −iΦ is continuous due to Corollary 1. Then we can take Hilbert space H to be (isomorphic to) the completion of Φ w.r.t. this norm in the spirit of original construction presented in [11]; see also [9] for another definition of continuous norm on inductive limit spaces. So we have proved the following proposition. Proposition 2. Consider the following data: • Separable infinite-dimensional Hilbert space H. • Strict inductive limit of a sequence of finite-dimensional Hilbert spaces. Then starting from one of these data we can construct another one in such a way that (Φ,H, Φ′) becomes a RHS.

Example 2. Let Hn be the space of polynomials on ξ ∈ R on degree ≤ n 2 and let H = L (R). Consider continuous embedding Hn → H : P (ξ) 7→ 2 e−x /4P (ξ), then due to the density theorem ([1], Theorem 4.5.1) we see ′ that Φ = lim→ Hn is dense in H, so (Φ,H, Φ ) is a RHS.

2 References

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Institute for Theoretical Physics, NSC Kharkov Institute of Physics and Technology, Akademicheskaia St. 1, 61108 Kharkov, Ukraine