Equicontinuity and Bilinear Mappings

A denotes a commutative topological ring with an identity element and A∗ denotes the multiplicative group of all invertible elements of A. All modules under consideration are unitary A-modules. E, F and G represent topological A-modules, M (resp. N ) represents a set of bounded subsets of E (resp. F), and Lsep(E,F; G) represents the A-module of all separately continuous A-bilinear mappings from E × F into G.

Let X ⊂ Lsep(E,F; G). (B1) X is point wise bounded if, for each (x, y) ∈ E × F, the set Is bounded in G. (B2)(a) X is M-uniformly bounded if, for each y ∈ F and for each B ∈ M, the set Is bounded in G. (B2)(b) X is N -uniformly bounded if, for each x ∈ E and for each C ∈ N , the set Is bounded in G. (B3) X is (M, N )-uniformly bounded if X is M-uniformly bounded and N -uniformly bounded. (B4) X is (M×N )-uniformly bounded if, for each B ∈ M and for each C ∈ N , the set Is bounded in G.

Since every subset of a topological module over a discrete ring is necessarily bounded, there is no interest in the study of the notions just defined when A is a discrete ring.

(a) If then (B2)(a) implies (B1) (resp. (B2)(b) implies (B1)). (b) If then (B4) implies (B2)(a) (resp. (B4) implies (B2)(b)). In particular, if then (B4) implies (B3). There are examples showing that the reverse implications in Remark are not valid in general; see

Let X ⊂ Lsep(E,F; G). (E1)(a) X is left equicontinuous if, for each y ∈ F, the set Is equicontinuous (E1)(b) X is right equicontinuous if, for each x ∈ E, the set. Is equicontinuous. (E2)(a) X is N -equihypocontinuous if, for each C ∈ N , the set Is equicontinuous. (E2)(b) X is M-equihypocontinuous if, for each B ∈ M, the set Is equicontinuous. (E3) X is (M, N)-equihypocontinuous if X is M-equihypocontinuous and N -equihypocontinuous. (E4) X is equicontinuous if, for each (x, y) ∈ E×F and for each neighborhood W of zero in G, there exist a neighborhood U of zero in E and a neighborhood V of zero in F such that the relations u ∈ X, x0 ∈ U, y0 ∈ V imply u(x0 +x, y0 +y)−u(x, y) ∈ W.

If then (E2)(a) implies (E1)(a) (resp. (E2)(b) implies (E1)(b)). In particular, ifthen (E3) implies (E1)(a) and (E1)(b).

Suppose that the product of any neighborhood of zero in A by any neighborhood of zero in E (resp. F) is a neighborhood of zero in E (resp. F). Then (E4) equicontinuous, let C ∈ N , and assume that the product of any neighborhood of zero in A by any neighborhood of zero in E is a neighborhood of zero in E. Given an arbitrary neighborhood W of zero in G, there are a neighborhood U of zero in E and a neighborhood V of zero in F such that X(U × V ) ⊂ W. By the boundedness of C, there exists a neighborhood L of zero in A such that LC ⊂ V . Thus Since, by assumption, LU is a neighborhood of zero in E, we have just verified that the set Is equicontinuous. Therefore X is N -equihypocontinuous. By interchanging the roles of E and F, we conclude that the other assertion is also true. There are examples showing that the reverse implications in Remarks are not valid in general; see [2]. In the sequel we shall see conditions under which such reverse implications hold.

The idea of a together consistent bilinear mapping between topological vectors spaces has been considered broadly; for instance, [13, 14, 15] for more data. Specifically, when we manage the normed spaces structure, these mappings convey limited sets (concerning the item topology) to limited sets. Then again, tensor items are a productive and helpful device in changing over a bilinear mapping to a direct administrator in any setting; for instance, the projective tensor item for normed spaces and the Fremlin projective tensor items for vector grids and Banach cross sections [8, 9]. In a topological vector space setting, we can consider two distinctive non-identical approaches to characterize a limited bilinear mapping. For reasons unknown, these parts of boundedness are as it were "middle of the road" thoughts of a mutually persistent one. Then again, various sorts of limited straight administrators between topological vector spaces and a portion of their properties have been examined [16, 18]. In this area, by utilizing the idea of projective tensor item between locally raised spaces, we show that, it could be said, various thoughts of a limited bilinear mapping match with various parts of a limited administrator. We demonstrate that for two limited straight administrators, the tensor item administrator likewise has a similar boundedness property, also.

Let X, Y , and Z be topological vector spaces. A bilinear mapping ζ : X × Y → Z is said to be: (i) n-bounded if there exist some zero neighborhoods U ⊆ X and V ⊆ Y such that ζ (U × V) is bounded in Z; (ii) b-bounded if for any bounded sets B1 ⊆ X and B2 ⊆ Y , ζ(B1 × B2) is bounded in Z We first show that these concepts of bounded bilinear mappings are not equivalent.

Let X = R N be the space of every genuine arrangement with the Tychonoff item topology. Consider the bilinear mapping ζ : X × X → X characterized by ζ(x, y) = xy where x = (xi), y = (yi) besides, the thing is point shrewd. It is successfully affirmed that ζ is b-constrained; anyway since X isn't secretly restricted, it can't be a n-restricted bilinear mapping. It isn't difficult to see that every n-constrained bilinear mapping is commonly incessant and each together relentless bilinear mapping is b-restricted, with the objective that these thoughts of constrained bilinear mappings are related to together steady bilinear mappings. Note that a b-constrained bilinear mapping needs not be commonly steady, even autonomously tenacious; by a freely incessant bilinear mapping; we mean one which is relentless in all of its portions. Consider the going with model

Let X be the space C[0, 1], consisting of all real continuous functions on [0, 1]. Suppose η1 is the topology generated by the seminorms px(f) = |f(x)|, for each x ∈ [0, 1], and η2 is the topology induced by the metric defined via the formulae

Consider the bilinear mapping ζ : (X, η1)×(X, η1) → (X, η2) defined by ζ(f, g) = fg. It is easy to show that ζ is a b-bounded bilinear mapping. But is it not even separately continuous; for example the mapping g = 1X, the identity operator from (X, η1) into (X, η2), is not continuous. To see this, suppose

V is a zero neighborhood in (X, η2). If the identity operator is continuous, there should be a zero For each subinterval [xi , xi+1], consider positive reals αi and αi+1 such that xi < αi< αi+1 < xi+1. For an n ∈ N, Define, Now consider the continuous function f on [0, 1] defined byobviouslyWe can choose n ∈ N and β in such a way thatThus, This finishes the case. In what pursues, by utilizing the idea of the projective tensor result of locally arched spaces, we are demonstrating that these ideas of limited bilinear mappings are, truth be told, the various sorts of limited administrators characterized on a locally curved topological vector space. Review that if U and V are zero neighborhoods for locally raised spaces X and Y, separately, at that point co(U ⊗ V ) is a run of the mill zero neighborhood for the locally curved space X⊗πY .

Let X, Y and Z be locally convex vector spaces and θ : X × Y → X⊗πY be the canonical bilinear mapping. If ϕ : X × Y → Z is an nbounded bilinear mapping, there exists an nb-bounded operator T : X⊗πY → Z such that T ◦ θ = ϕ.

By [15], there is a straight mapping T : X⊗πY → Z with the end goal that T ◦ θ = ϕ. Subsequently, it is sufficient to show that T is nb-limited. Since ϕ is nbounded, there are zero neighborhoods U ⊆ X and V ⊆ Y with the end goal that ϕ(U × V ) is limited in Z. Give W a chance to be a subjective zero neighborhood in Z. There is r > 0 with ϕ(U × V ) ⊆ rW. It isn't difficult to show that T (U ⊗ V ) ⊆ rW, so T (co(U ⊗ V )) ⊆ rW. However, by the reality referenced before this recommendation, co(U ⊗V ) is a zero neighborhood in the space X⊗πY . This finishes the verification

: X × Y → X⊗πY be the canonical bilinear mapping. If ϕ : X ×Y → Z is a b-bounded bilinear mapping, there exists a bb-bounded operator T : X⊗πY → Z such that T ◦ θ = ϕ.

As in the proof of the previous theorem, the existence of the linear mapping T : X⊗πY → Z such that T ◦ θ = ϕ follows by [15]. We prove that the linear mapping T is bb-bounded. Consider a bounded set B ⊆ X⊗πY . There exist bounded sets B1⊆ X and B2⊆ Y such that B ⊆ B1⊗ B2. To see this, put

It is not difficult to see that B1 and B2 are bounded in X and Y , respectively, and B ⊆ B1⊗ B2. Also, since θ is jointly continuous, B1⊗ B2 is also bounded in X⊗πY . Thus, from the inclusion

And using the fact that ϕ is a b-bounded bilinear mapping, it follows that T is a bb-bounded linear operator. This concludes the claim and completes the proof of the proposition

Note that the similar result for jointly continuous bilinear mappings between locally convex spaces is known and commonly can be found in the contexts concerning topological vector spaces [15]. We are going now to investigate whether or not the tensor product of two operators preserves different kinds of bounded operators between topological vector spaces. The response is affirmative. Recall that for vector spaces X, Y , Z, and W, and linear operators T : X → Y , S : Z → W, by the tensor product of T and S, we mean the unique linear operator T ⊗S : X ⊗Z → Y ⊗W defined via the formulae

One may consult [14] for a comprehensive study regarding the tensor product operators.

Let X, Y , Z, and W be locally convex spaces, and T : X → Y and S : Z → W be nb-bounded linear operators. Then the tensor product operator T ⊗ S : X⊗πZ → Y ⊗πW is nb-bounded. such that T (U) and S(V ) are bounded subsets of Y and W, respectively. Let O1 ⊆ Y and O2⊆ W be two arbitrary zero neighborhoods. There exist positive reals α and β with T (U) ⊆ αO1 and S(V ) ⊆ βO2. Then

So that This is the desired result.

Suppose X, Y , Z, and W are locally convex spaces, and T : X → Y and S : Z → W are bb-bounded linear operators. Then the tensor product operator T ⊗ S : X⊗πZ → Y ⊗πW is also bb-bounded

Fix a bounded set B ⊆ X⊗πZ. By the argument used in Proposition, there are bounded sets B1 ⊆ X and B2⊆ Z with B ⊆ B1⊗B2. Let O1⊆ Y and O2⊆ W be two arbitrary zero neighborhoods. There are positive reals α and β such that T (B1) ⊆ αO1 and S(B2) ⊆ βO2. Therefore,

Hence As required

Suppose X, Y , Z, and W are locally convex spaces, and T : X → Y and S : Z → W are continuous linear operators. Then the tensor product operator T ⊗ S : X⊗πZ → Y ⊗πW is jointly continuous.

Let O1⊆ Y and O2⊆ Z be two arbitrary zero neighborhoods. There exist zero neighborhoods U ⊆ X and V ⊆ Z such that T (U) ⊆ O1 and S(V ) ⊆ O2. It follows

So that

As claimed.

It should be mentioned that this paper was written under the influence of where the same notions for sets of separately continuous bilinear mappings between topological vector spaces have been studied. Throughout this work, A denotes a commutative topological ring with an identity element and A∗ denotes the multiplicative group of all invertible elements of A. All modules under consideration are unitary A-modules. E, F and G represent topological A-modules, M (resp. N ) represents a set of bounded subsets of E (resp. F), and Lsep(E,F; G) represents the A-module of all separately continuous A-bilinear mappings from E × F into G.

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Research Scholar, P.G Department of Mathematics, Patna University, Patna, Bihar santoshrathore.kumar20@gmail.com