Nuclear Mappings and Quasi-Nuclear Mappings

Journal of Information and Computational Science ISSN: 1548-7741

NUCLEAR MAPPINGS AND QUASI-NUCLEAR MAPPINGS

Dr. Santosh Kumar1, Dr Raj Kumar2 1Faculty, Department of Mathematics, Patna Science College, Patna University, Patna, Bihar. E-Mail: [email protected] 2St Xavier’s College of Management and Technology, Digha Ashiyana Road, Patna, Bihar. E-Mail: [email protected]

ABSTRACT

In the present paper, we look at different approaches towards tensor products, which are known to coincide for Banach spaces, and give necessary and sufficient conditions for the dual spaces of the particular semi Banach spaces. A while later we consider the injective tensor standard as one especially well known model for standards on tensor results of semi Banach spaces. As we will see, there are three different descriptions of injective quasi-norms. This condition turns out to be theinsignificant necessity. Additionally, we study augmentations of the old style injective and p-nuclear tensor standards to semi Banach spaces. Specifically, we give an adequate condition for the p-atomic semi standards to be crossnorms, which especially

applies to the case of weighted 푙푝-sequence spaces.

KEYWORDS- Algebraic Tensor Product, Quasi-Norm, P-Norm, Injective Tensor Norm, P- Nuclear Tensor Norm.

1. INTRODUCTION

In mathematics, a nuclear space is a topological vector space with various better properties of finite-dimensional vector spaces. The topology on them can be characterized by a family of seminorms whose unit balls decrease quickly in size. Vector spaces whose components are "smooth" in some sense will in general be atomic spaces; a run of the mill case of an atomic space is the arrangement of smooth capacities on a conservative manifold [1].

All limited dimensional vector spaces are nuclear (in light of the fact that each administrator on a limited dimensional vector space is nuclear). There are no Banach spaces that are nuclear,

Volume 10 Issue 7 - 2020 390 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

with the exception of the limited dimensional ones. By and by a kind of banter to this is regularly valid: in the event that a "normally happening" topological vector space isn't a Banach space, at that point there is a decent possibility that it is nuclear.

While for tensor results of Hilbert spaces, a significant number of these perspectives are surely known because of the unique properties of these Hilbert spaces, the hypothesis of tensor results of Banach spaces is unmistakably increasingly included. Be that as it may, in present day estimate hypothesis semi Banach spaces turned out to be progressively significant. In this setting they regularly show up while portraying purported estimate spaces, for example given a fixed estimation strategy one is keen on the assortment of all capacities with a given combination rate[2].

Definition 1: A nuclear space is a nearby convex topological vector space for instances any seminorm p we can get a larger seminorm q so as to the normal chart from Vq to Vp is nuclear.

Casually, this implies at whatever point we are given the unit wad of some seminorm, we can locate an "a lot littler" unit bundle of another seminorm inside it, or that any area of 0 contains an "a lot littler" neighborhood. It isn't important to check this condition for all seminorms p; it is adequate to check it for a lot of seminorms that produce the topology, as it were, a lot of seminorms that are a sub base for the topology.

Rather than utilizing subjective Banach spaces and nuclear administrators, we can give a definition as far as Hilbert spaces and follow class administrators, which are more obvious. (On Hilbert spaces nuclear administrators are frequently called follow class administrators.)We will state that seminorm p is a Hilbert seminorm if Vp is a Hilbert space, or equivalently if p comes from a sesquilinear optimistic semi definite form on V.

Definition 2: A nuclear space is a topological vector space through topology definite by a family of Hilbert seminorms, such that for some Hilbert seminorm p we can discover a better

Hilbert seminorm q so that the ordinary chart from Vq to Vp is trace class.

Some authors prefer to use Hilbert–Schmidt operators rather than follow class administrators. This has little effect, in light of the fact that any follow class administrator is Hilbert–Schmidt, and the result of two Hilbert–Schmidt administrators is of follow class.

Definition 3: An nuclear space is a topological vector space with a topology characterized by a group of Hilbert seminorms, to such an extent that for any Hilbert seminorm p we can find a larger Hilbert seminorm q so that the natural map from Vq to Vp is Hilbert–Schmidt.

On the off chance that we are happy to utilize the idea of an nuclear administrator from a self- assertive locally raised topological vector space to a Banach space, we can give shorter definitions as pursues:

Definition 4: A nuclear space is a locally convex topological vector space such that for any seminorm p the natural map from V to Vp is nuclear.

Volume 10 Issue 7 - 2020 391 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

Definition 5: A nuclear space is a locally arched topological vector space with the end goal that any persistent straight guide to a Banach space is nuclear.

Grothendieck utilized a definition like the accompanying one:

Definition 6: A nuclear space in the vicinity arched topological vector space A with the end goal that for any locally curved topological vector space B the characteristic guide from the projective to the injective tensor item of A and B is an isomorphism [3].

2. TENSOR PRODUCTS OF (QUASI-)BANACH SPACE 2.1. ALGEBRAIC AND ANALYTIC DEFINITION

In algebra tensor product constructions are known for several different structures. The starting point for one possibility of an explicit construction for vector spaces X and Y (with respect to the same field; Thus we focus on real or complex vector spaces) is the free vector space F(X,Y) on X × Y , i.e. the set

Afterwards the algebraic tensor product X ⊗ Y is defined as the quotient space of F(X, Y) with respect to the subspace

In this way the canonical mapping (x, y)→ x ⊗ y from X × Y to X ⊗ Y becomes bilinear. The common practical explanatory approach for normed spaces X and Y is marginally different. Once more one starts with F(X, Y), but this times this space is equipped with the following 푛 ′ equivalence relation. We say 푓 = ∑푗=1 휆푖푥푗⨂푦푖 ∈ 퐹(푋, 푌) generates an operator 퐴푓: 푋 → Y by the determination

i.e. f and g generate the same operator from the dual space 푋′of X to Y . Of interest now is the quotient space 푋⨂퐴푌 = F(X, Y )/ +, which is found to coincide as a vector space with X ⊗ Y

Volume 10 Issue 7 - 2020 392 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

By this definition the connection with linear mappings from 푋′to Y is made obvious right from the beginning. This systematic approach applies to semi normed spaces also, however since the double space is potentially minor, this identicalness connection just as the separate remainder space may get inconsequential. To stay away from this, for example to guarantee the proportionality of the two methodologies, we need to force certain confinements on the semi normed spaces. This circumstance is explained by the accompanying hypothesis.

Theorem 1. Let X and Y be two quasi-normed spaces. Then it holds X ⊗ Y = X ⊗A Y if, and ′ only if, 푋 Separates the points in X, i.e. for every x ∈ X \ {0} there exists a functional 휑푥 ∈ ′ 푋 , such that 휑푥(x) = 0. A quasi-Banach space X with this property is said to have a separating dual.

Proof. So as to show the occurrence of the two spaces we need to show that U = V= {f∈F(X,

Y): 퐴푓 = 0 } grasp. The insertion U ⊂ V is apparent. For the repealen closure we comment that the situation on 푋′ is corresponding to Ax⊗y = 0 for every x = 0 and y = 0. To show now V ⊂ U, we show instead, that from f∈ U follows f ∈ V. We shall utilize the fact that for each f 푛 ∈ U there survive an (algebraically) corresponding illustration푓 = ∑푖=1 푥푖 ⨂ 푦푖, where {푥1, . . 1 푛 . , 푥푛} ⊂ X and {푦 , . . . , 푦 } ⊂ Y are linearly autonomous (this can be seen analogously to 1 푛 [Lemma 1.1]). The linearity of f) →퐴푓, the linear independency of {푦 , . . . , 푦 } and the ′ assumption for 푋 (applied to the vectors 푥푖 = 0 now yield 퐴푓 = 0. In a similar way, we can also consider operators.

We then observe that

Holds for all f, g ∈ X ⊗ Y if, and only if, 푋′ and 푌′are separating. This follows from F(X, Y) ≅ F(Y, X) and 푋⨂퐴푌 ≅ X ⊗ Y ≅ Y ⊗ X ≅ Y ⨂퐴푋 , where the isomorphism is provided by the canonical identification X ⊗ Y)→ Y ⊗ X, x ∈ X, y ∈ Y (for the logarithmic tensor item this is in every case genuine, and the suspicions guarantee this reaches out to the separate (useful diagnostic) comparability relations). Because of this perception we hereafter consistently expect that X and Y both have isolating duals (without in every case unequivocally referencing it).

REMARK1. If one is merely involved in equipping the algebraic tensor product X ⊗ Y of general topological vector spaces with just some topological structure, then one does not need information on the respective dual spaces. For example, if X and Y are equipped with topologies which are Hausdorff, then there exists a topology on X ⊗ Y which is Hausdorff as well. For more information on such topological issues, we refer to Turpin [4, 5] and Waelbroeck [6].

Volume 10 Issue 7 - 2020 393 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

3. INJECTIVE QUASI-NORMS

3.1 ASSOCIATED NORMS AND BANACH ENVELOPES

Tensor products arise quite naturally in the study of bilinear forms and their relation to linear operators. Hence it is no great surprise that also the Hahn-Banach extension theorem is frequently used in proofs, in particular the well-known identity

Which is valid for any (real or complex) normed vector space V . It is well-known that the Hahn-Banach extension theorem generally fails to be true for quasi-Banach spaces. Hence a natural question when trying to transfer proofs from Banach spaces to quasi-Banach spaces is whether at least the identity remains valid. The answer is given by the following lemma.

LEMMA 3. Let X be a quasi-Banach space. Then we define

Moreover, the completion of X with respect to the associated norm is called the Banach envelope and will be denoted by 푋퐸. In particular, it follows 푋′ =(푋퐸)′. For further details and

references we refer to [7]. Since푙푝, p < 1, is known not to be locally convex and hence also not normable, both conditions in (ii) and (iii) would exclude these spaces. Thus they are far too restrictive to be feasible. This means many proofs for tensor products of Banach spaces relying on equation (1) cannot be carried over without changes to the quasi-Banach case, at least without exceptional additional assumptions on the spaces involved. Before we return to tensor product spaces, we shall recall a last well-known notion for quasi-Banach spaces.

Definition 7. Let 0 < p ≤ 1, and let X be a quasi-Banach space. Then X is called a p-Banach space and its quasi norm p-norm, respectively, if

It is clear, that every Banach space is a 1-Banach space and every norm is a 1-norm. Furthermore, it can be shown that for every quasi-Banach space (푋, ‖. ‖) there exists a p ∈ (0,

Volume 10 Issue 7 - 2020 394 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

1] and a p-norm ‖. ‖∗ on X, which is equivalent to ‖. ‖ i.e. (푋, ‖. ‖) is a p-Banach space. We refer to [17] for details and further references.

3.2 풑-NUCLEAR-NORMS FOR QUASI-BANACH SPACES

For Banach spaces X and Y , the p-nuclear tensor norm αp(·, X, Y ) is well-known, where 1 < p < ∞, see e.g. [8]. Its definition can be extended to values p ≤ 1 in different ways, which are seen to coincide for Banach spaces (for completeness we shall repeat those arguments below). However, they are generally different for quasi-Banach spaces.

Definition8. Let X and Y be quasi-Banach spaces, and let 푓휖푋 ⨂ 푌.

(i) Let 0 < p ≤ 1. Then we define the p-nuclear norm 훾푝 by

1 1 (ii) Now let 0 < 푝 ≤ ∞ and = (1 − ) Then we define 푞 푝

with the usual modification in case q = ∞(i. e. p ≤ 1).

(iii) Finally, let again 0 < 푝 ≤ ∞. Then we define

푛 푛 푛 Where 푙푝 p is the vector space ℂ , equipped with the usual (quasi-)norm ‖휆|푙푝 ‖ = 1 푛 푝 푝 (∑푗=0|휆푖| )

The version (i) was used already by Grothendieck in [9]. It can be shown that for Banach spaces

the projective norm 훾1 = γ is always equal to the 1-nuclear norm 훼1 as defined in [8] (which justifies the above notion in case 푝 = 1). On the other hand, (ii) and (iii) are more immediate extensions of the usual formulations of the p-nuclear tensor norm.

4. CROSSNORM-PROPERTIES

In this section we shall investigate the 푝 -nuclear tensor norms from Definition 4 regarding

possible crossnorm-properties. To begin with, we observe that 훼푝 generally is neither a crossnorm, nor equivalent to one. Instead, we have the following assertion.

Volume 10 Issue 7 - 2020 395 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

LEMMA 1. Let푋 and 푌be quasi-Banach spaces, and let 0 < 푝 ≤ 1 Then it holds

And furthermore

Proof. The first assertion can already be found in the proof of Lemma 9, where we have shown

Applying this to 푧 = 푥⨂푦, we find

Since the reverse estimate simply follows by inserting 푋 ⊗ Yinto the definition of훼푝, the proof

is complete. From (10) we immediately obtain results for 푋⨂훾푝Y similarly as in Lemma 4

LEMMA 2. Let X and Y be p-Banach spaces, where 0 < p ≤ 1. Then it holds

In particular, we find

Theorem 1.Let X be a quasi-Banach space with separating dual, and let Y be another quasiBanach space. Assume that for arbitrary x, x1, . .. , xn ∈ X and y, y1, . . . , yn ∈ Y , such that x⊗y .= 0 and

The following condition is fulfilled:

Volume 10 Issue 7 - 2020 396 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

∗ Where X∗ denotes the algebraic dual 푋 of X, and 퐶1 does not depend on x, x1, . . . , xn ∈ X and n ∈ N. Then it holds

푛 푝 Note that the restriction on x, 푥1, . . . , xn ∈ X implies that ∑푖=0|휑(푥푖)| is non-vanishing for ∗ all functional ϕ ∈푋 \ {0}. Clearly, we always have 퐶1 ≤ 1 (take n = 1 andx = 푥1). The proof is based on an argument by Nitsche [11]. We further remark that for Banach spaces and 푙푝- ∗ ′ spaces we have 퐶1 = 1 (even upon replacing 푋 by 푋 ).

Proof. Let 푥, 푥푖, … . , 푥푛 ∈ 푋 and 푦, 푦푖, … . , 푦푛 ∈ 푌 be given as required. Moreover, consider an φ(xi) arbitrary functional ϕ ∈ X∗ \ {0}. We put 휂 = Then it follows 푖 M

Taking the supremum over all such functional and afterwards taking the infimum over all equivalent representations of x ⊗ y yields the claimed result. To finish this subsection we adapt another result of Nitsche, giving an example of spaces satisfying condition (12). We suppose

• X is a p-Banach space,

• X contains a Schauder basis 퐵 = (푏푖)푖휖퐼, where I is a countable index set,

• The mapping 퐽: 퐹 → (휆_푖 (푓)‖푏_푖 |푋‖ )푖휖퐼is bounded from X to 푙푝(I).

Theorem 3. Under these assumptions it holds

−1 Where the constant 퐶0 is given by 퐶0 = ‖퐽|ℒ(푋, 푙푝(퐼))‖

Proof. We shall show that every space X with a basis as assumed satisfies the condition (12) 0 푖 with 퐶1 = 퐶푝 . Throughout the proof (휆푖)푖휖퐼 ⊂ 푋 denotes the system of the correponding

coefficient functional. Now let 푥, 푥1 … , 푥푛 ∈ X be given as in the last theorem, which particularly implies x = 0 and

Volume 10 Issue 7 - 2020 397 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

The assumption on the mapping J then yields

Now consider the set

By construction we find for all ϕ ∈ Φ

Now assume that there is no 휑 ∈ Φ, such that 휑(x) ≥ 퐶0‖푥|푋‖(observe that 휑(x) is real for all 휑 ∈ Φ). This implies 휑푖(x) ≥ 퐶0‖푥|푋‖for all i ∈ I and hence (summing up over i ∈ I)

Moreover we find from the p-triangle inequality

Altogether we find

−1 Hence in case 퐶0 = ‖퐽|ℒ(푋, 푙푝(퐼))‖ this is a contradiction, thus a functional ϕ ∈ Φ with 0 휑푖(x) ≥ 퐶0‖푥|푋‖ does exist. This functional then yields condition (19) with 퐶1 = 퐶푝 , what proves the claim.

CONCLUSION

The aim of the present paper, investigate under which conditions on quasi-Banach spaces one can carry over at least some of the important results for their respective tensor products. As regarding topological issues like existence of Hausdorff topologies on the (algebraic) tensor

Volume 10 Issue 7 - 2020 398 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741

product of general topological vector spaces this has been done by other researchers, but little seems to be known concerning tensor quasi-norms. We look at different approaches towards tensor products, which are known to coincide for Banach spaces. There are three distinct renditions of injective semi standards. Regarding these semi standards we will exhibit that likewise for other fundamental properties of such tensor standards a few non-comparable augmentations to semi Banach spaces occur. In the last section we apply these considerations

first to weighted 푙푝 -sequence spaces and a short time later to work spaces of Sobolev-and Besov-type. These spaces are notable to permit a portrayal as far as wavelet bases and related succession spaces, and we will think about the reliance of the standard of the comparing isomorphism on the component of the hidden area.

REFERENCES

[1]. Grothendieck, Alexandre (1955). "Produits tensoriels topologiques et espaces nucléaires". Mem. Am. Math. Soc. 16. [2]. Gel'fand, I. M.; Vilenkin, N. Ya. (1964). "Generalized Functions – vol. 4: Applications of harmonic analysis". OCLC 310816279. [3]. Takeyuki Hida and Si Si, Lectures on white noise functionals, World Scientific Publishing, 2008. ISBN 978-981-256-052-0 [4]. P. Turpin, Repr´esentation fonctionelle des espaces vectorielles topologiques. Studia Math. 73 (1982), 1–10. [5]. P. Turpin, Produits tensorielle d’espaces vectoriels topologiques. Bull. Soc. Math. France 110 (1982), 3–13. [6]. L. Waelbroeck, The tensor product of a locally pseudo-convex and a nuclear space. Studia Math. 38 (1970), 101–104. [7]. N. J. Kalton, Quasi-Banach spaces. In: Handbook of the Geometry of Banach spaces, Volume 2. Edited by W.B. Johnson, J. Lindenstrauss. North Holland, Amsterdam, 2003, 1099– 1130 [8]. W. A. Light, E. W. Cheney, Approximation theory in tensor product spaces. Lecture Notes in Math. 1169, Springer, Berlin, 1985.

[9]. A. Grothendieck, R´esum´e de la th´eorie m´etrique des produits tensoriels topologiques. Bol. Soc. Mat. S˜ao Paulo 8 (1956), 1–79.

[10]. A. Pietsch, History of Banach spaces and linear operators. Birkh¨auser, Boston, 2007

[11]. P.-A. Nitsche, Best N-term approximation spaces for tensor product wavelet bases. Constr. Approx. 24 (2006), 49–70.

[12]. W. Sickel, T. Ullrich, Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross. J. Approx. Theory 161 (2009), 748–786.

Volume 10 Issue 7 - 2020 399 www.joics.org