A New Hypothesis Of Projective Tensor Product And It’s Consequences

A New Hypothesis Of Projective Tensor Product And Its Consequences

Mr. Ajay Kumar1, Prof. Seema Sharma2 Dr. Sushil Kumar Jamariar3

Department of Mathematics

1. Dr. C.V. RAMAN UNIVERSITY, Bhagwanpur, Vaishali , Bihar,

2. Dr. C.V. RAMAN UNIVERSITY, Khandwa, (M.P)

3. Dr. C.V. RAMAN UNIVERSITY, Bhagwanpur, Vaishali , Bihar,

ABSTRACT

This Paper Presents the Study of the Projective Tensor Product and its consequences by defining the projective

topology on Locally convex spaces E and F; U & V be the closed absolutely convex neighborhoods of Q in E and F respectively , forming the set (UV) = absolutely convex hull of UV in EF, it is proved in this paper that the projective topology is the finest locally convex topology on EF for which the Canonical mapping : E X F EF is continuous.

Tensor products are used to describe systems consisting of multiple subsystems. Each systems are described by a vector in a

Hilbert Space.

KEYWORDS : Projective Topology, Locally convex spaces, Convex hull, Canonical mapping, Automorphism, Canonical Isomorphism.

INTRODUCTION:

The strongest locally convex topological vector space (TVS) topology on XY the tensor product of two locally Convex TVSs making the Cononical map : X × Y X Y defined by sending (x,y) X × Y to XY Continous is called projective topology or Topology . When XY is equipiped with this topology then it is denoted by XY and called

the projective tensor product of X and Y. Halub (1) and Kothe (2,3) are the pioneer worker of the present area. In fact the present

work is the extension of work done by Tomiyama (6), Studied analytically about projective Tensor Product.

Here, we use the following definitions and fundamental ideas:

Definition – I

If V and W be vectors space of finite dimension then V W is fnite diminsional and its dimension is product of the dimensions of V and W. This result from the fact that V W is formed by taking all Tansor Products of basis element

of V and basis element of W.

Definition II

Let E and F be locally convex spaces , and let U & V be the closed absolutely convex neighborhoods of O in E and F

respectively, forming the set (UV) = absolutely convex, hull of UV in EF , (EF is denoted as tensorial product of E & F).

Definition II

If {U} and {V} are neighborhood bases in E and F respectively with U,V closed absolutely then the family { (UV)} is a neighbourhood basis of a locally convex topology on EF

This topology is called the projective topology on EF and is denoted as E.

Properties

1. Associativity as a Vector Space operation

   (UV) W U(VW)

   Where U,V,W be canonical isomorphism that maps

   (UV) W to U(VW)

   This follows omitting parenthesis in the tensor product of more than two vector space or vectors.

2. Commutativity as a vector space operation

V W = WV

Where V & W is commutative in the sense that is a canonical isomorphism.

That maps V W = WV

In fact V= W the tensor product of vectors is not commutative that is V W WV

In general X Y Y X from V V to itself induces a linear auto morphism that is called braiding map.

Proposition 1 : Let p(x) and q(y) be the semi-norms defined by U and V respectively. The set (UV) is absorbing and thus defines a semi-norm . The semi-norm of (UV) is given by

pq (-Z) = in f1.p(xi)q(yi) ————- (1) Where the infimum is taken over all representations

-Z = .xi yi in E F.

Proof : First we show (UV) is absorbing . Let

-Z = .xi yi be an element of E F. we observe that U if p(Xi) 0

And yi

(yi)

V if q(Yi) 0

Also p(Xi) = 0 iff pixi U all p > 0 and

q(Yi) = 0 iff qiyi V all p > 0 So we write

-Z = .xi yi

= p(X ) q(Y ) [

i i

()

]

()

+ q.(Yk) [

]

()

+ p.(Xj ) [

]

()

m

+ 2

. [

+ ]

In each of the four terms in the sum representing -Zi the quantity in the brackets [ ] is in (UV). Given > 0 , we may

choose sufficiently small so that

(*) : – Z (.p(xi)q(yi) + ) (UV) So (UV) is absorbing.

Now (UV) is absorbing convex also , so it defines semi- norm (-Z) on EFew now show (-Z) = p q(-

Z)

1. (-Z) p (-Z) , (-Z) is defined by

   (-Z) = inf , -z (UV) , by ( *) above

   (-Z) inf .p(xi)q(yi) + = p q(-Z) + , arbitrary yields (-Z) p q(-Z)

2. p q(-Z) (-Z) suppose -Z (UV), Then

-Z = . ( ) with p () 1 , q ( ) 1, .I I and 0 . For this particular

representation of Z , we see

.p(xk) q(yk) = .I I So. p q(-Z) =inf .p(xi)q(yi) .

This is true for every with Z (UV). Thus p q(-Z) inf , -z (UV)

= (-Z)

This completes the proof

Proposition 2 : The projective tensor product EF of two normed space E, p and F. q is a normed space with norm pq

If E and F are metrizable locally convex spaces with semi norms p1 p2 . and q1 q2 .. respectively, then E

Fis metrizable with defining semi- norms p1 q1 p2 q2

Proof: Follows immediately from Proposition 1 and definition 2.

MAIN RESULT

Theorem: The projective topology is the finest locally convex topology on EF for which the canonical map : E X F

EF is continuous.

Proof: is continuous with respect to since : U X V UV (UV)Now let be any topology on EF for which is continuous and let W be an absolutely convex closed neighborhood of 0. Then there exist U, V with (U x V) – U V W

.Since W is absolutely convex, (UV) W. So is finer than

This completes the proof of the hypothesis.

Acknowledgement

The authors are Thankful to Prof (Dr.) Basant Singh, Ex. Head of the Department of Mathematics. R.N.T.U Bhopal and Prof (Dr.) Dharmendra Kumar Singh, Dean Academic , Dr. C. V. R.U Vaishali ,Bihar, India . Thanks to library and its in charge , Dr. C .V . Raman University ,Vaishali, Bihar, India. For extending all facilities in the completion of the present research work

REFERENCES:

[1] Halub, JR. (1970) : Tensor Product Mapping Math: Ann., Vol. 188, pp01-12 Kothe, G. (1969): Topological Vector Spaces, Springer Verlage, 1, New York.

[2] Kothe, G. (1979) Topological Vector Spaces, Springer Verlag. II. New York

[3] Srivastava, UK, Kumar, S, and Singh, TN (2011)J.P.AS. (Mathematical Science), Vol 17, pp 188-191.

[4] Srivastava, UK, Kumar, N., Kumar, S., and Singh, TN. (2012) JP.A.S. (Mathematical Science), Vol 18, pp189-191 [5] Tomiyama, J. (1971) Tensor Products and Projection of Norms. One in VON -NEUMANN Algebras.

[6] Treves Fran Cars (2006) [1967] [7] Topological vector space, Distribution and kernel Mineola, NY Dover Publication ISBN 978-0-486 -45352-1. [8] Grillet Pierre (2007) Abstract Algebra springer science, business Media, LLC ISBN -978 -0387715674.

Gowers, Timothy(2007) How to lose your fear of tensor product Archived from the original (on 23 July 2021).