The following will be discussed: â¢ The Identity tensor â¢ Transpose of a tensor â¢ Trace of a tensor â¢ Norm of a tensor â¢ Determinant of a tensor â¢ Inverse of a tensor Introduction, uniqueness of tensor products x2.  N. Bourbaki, "Elements of mathematics. The correct or consistent approach of calculating the cross product vector from the tensor (a b) ij is the so called index contraction (a b) i = 1 2 (a jb k a kb j) ijk = 1 (a b) jk ijk (11) proof (a b) i = 1 2 c jk ijk = c i = 1 2 a jb k ijk 1 2 b ja k ijk = 1 2 (a b) i 1 2 (b a) = (a b) i In 4 dimensions, the cross product tensor â¦ properties of second order tensors, which play important roles in tensor analysis. Tensor products rst arose for vector spaces, and this is the only setting where they occur in physics and engineering, so weâll describe tensor products of vector spaces rst. Proof Both satisfy the Universal Property of the 2-Multi-Tensor Product. (A+B) T =A T +B T, the transpose of a sum is the sum of transposes. They are a good example of the phenomenon discussed in my page about definitions : exactly how they are defined is not important: what matters is the properties they have. %��������� In view of #1, we write for the generators in both (isomorphic) modules, as an “abuse of notation”. • A useful identity: ε ijkε ilm = δ jlδ km −δ jmδ kl. (Recall that a bilinear map is a function that is separately linear in each of its arguments.) Theorem 1. That this is a nice operation will follow from our properties of tensor products. According to the definition of the Kronecker product and the matrix multiplication, we have From Theorem 1, we have the following corollary. But in Vakil's Rising Sea, he asks one to prove this without knowing hom-tensor adjoontness. This construction often come across as scary and mysterious, but I hope to shine a little light and dispel a little fear. For -modules, 4. 1. G. Emmanuele and W. Hensgen, “Property (V) of Pelczyński in projective tensor products,” Proceedings of the Royal Irish Academy A: Mathematical and … Proof. Consider the universal property of Deï¬nition5.2(b) for the ï¬rst tensor product: as t 2: M N !T 2 is bilinear, there is a unique morphism j : T 1!T 2 with t 2 = j t 1. Proof: The basic idea of the proof is as follows: [(I m ⊗A)+(B ⊗I n)](z⊗x)= (z⊗Ax)+(Bz⊗x) = (z⊗λx)+(µz⊗x) = (λ+µ)(z⊗x). Let N0be a submodule of N. Then one has a canonical isomorphism (N=N0) M= N Many of the concepts will be familiar from Linear Algebra and Matrices. Tensor-product spaces â¢The most general form of an operator in H 12 is: âHere |m,nã may or may not be a tensor product state. But a better proof uses the fact that the tensor product represents Bilinear maps. What has all this to do with tensor products? Two techniques are relevant here: on one hand we may gain insight in the tensor product by using its universal property; on the other hand, as the next proof shows there is an explicit (but rather abstract!) It was shown in the proof of Lemma 1 that kT#k 6 kTk. The typical proof of the this by the hom-tensor adjoint thing. Indeed, the de nition of a tensor product demands that, given the bilinear map ˝: M N! Proof: The module axioms give us a surjective bilinear map T: R×M→ M given by T(r,m) = rm. In the context of vector spaces, the tensor product ⊗ and the associated bilinear map : × → ⊗ are characterized up to isomorphism by a universal property regarding bilinear maps. The result that both the inner and outer products of two tensors transform as tensors of the appropriate order is known as the product … Now is the time to admit that I have already defined tensor products - in two different ways. De nition 3.1.2 Let T be a linear vector space over . In this discussion, we'll assume VV and WW are finite dimensional vector spaces. Hence kTk = k(T#)#k 6 kT#k 6 kTk. 5. Then pX;bqsatis es the universal property of the tensor product: construct a factorization in (20.3) using V1 V2and then restrict to the subspace X. construction of the tensor product that we may use to deduce information. Let and ; then Proof. Similar results as above hold for -modules . On some properties of tensor product of operators 5143 3. We say that T satis es the char-acteristic property of the tensor product (with respect to V and W) if there is a bilinear map h: V W! Properties of the Kronecker Product 141 Theorem 13.7. Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor. Comments . (The simplest case is $\mathbb F_4 \otimes_{\mathbb F_2} \mathbb F_8=\mathbb F_{64}$.) Moreover, if $(T, g)$ and $(T', g')$ are two pairs with this property, then there exists a unique isomorphism $j : T \to T'$ such that $j\circ g = g'$. Similar results as above hold for -modules . Do a uniqueness argument. 1.1 Examples of Tensors . The correct or consistent approach of calculating the cross product vector from the tensor (a b) ij is the so called index contraction (a b) i = 1 2 (a jb k a kb j) ijk = 1 (a b) jk ijk (11) proof (a b) i = 1 2 c jk ijk = c i = 1 2 a jb k ijk 1 2 b ja k ijk = 1 2 (a b) i 1 2 (b a) = (a b) i In 4 dimensions, the cross product tensor … 1. BASIC PROPERTIES OF TENSORS . properties of second order tensors, which play important roles in tensor analysis. Let's try to make new, third vector out of vv and ww. We also study the determinants of tensors after two types of transposes. The Properties of the Mixed Products . Proof: First, we show that for a tensor product ˝: M N! 5 0 obj ���B�[M���I(!�DBV�&>*�O��x��e��^F��b�q����_�/��س��Mݶ�FJ�e��F�ѕn�8�;��k� Introduction 1.1. an open source textbook and reference work on algebraic geometry Algebra: Algebraic structures. Comments . We then list many of its properties without proof in Section 2.1, and conclude with some of its applications in Section 2.2. Notation: 6. This section discusses the properties based on the mixed products theorem [6, 33, 34]. In general, if A and B have Jordan form decompositions given by P−1AP = J A and Q−1BQ= J B, respectively, then [(Q ⊗ I $\endgroup$ â Georges Elencwajg Nov 28 â¦ Next we observed that the uniqueness of the tensor product is also built into the de nition of the tensor product. (A×B) , where we’ve used the properties of ε ijk to prove a relation among triple products with the vectors in a diﬀerent order. Therefore, the transpose of a tensor product of two matrices is equal to the tensor product of the transposed â¦ 1.1 Examples of Tensors . Taking tensor products of wavelets on [0, 1] immediately yields biorthogonal wavelet bases on the unit d-cube := [0, l] d with analogues of (3.2.3), (3.2.4), (3.2.5).One can push this line a little further in the following direction. 3. Let N0be a submodule of N. Then one has a canonical isomorphism (N=N0) M= N 2. $$\,$$ To be equal, two matrices should have the same sizes and the same corresponding entries. In view of #1, we write for the generators in both (isomorphic) modules, as an âabuse of notationâ. If A2M m;k and B2M n;â, then A Bis the block matrix with m k blocks of size n âand where the i;jblock is a i;jB. The gradient of a vector field is a good example of a second-order tensor. Matrix products: M m k M k n!M m n Note that the three vector spaces involved aren’t necessarily the same. x���n$ɑ���!�% cb��(�$@Lp �l6���j�����e�����ca�*2�����c�X����l��m�����f��61~w���w�z{�m������]״uW_}��\o��t���������z����Z_�8@��+Q0���a�9��;�f��N�/���_P����Qlh�3���٧�r�3x��zw�S���%��9.۾9t��WжM�1�m��Bm�jLG�q����E�6�͔���IH_��獡�M&� �aW�9���i4���[���աP�է�)^����M��;=!w�A�v�fo�M�O2���vGIp�%�� 0�������˱�̙�zQ�G��z�����f�Z0*i�0�g�r��Lo�bt~�m3 ��!a�w/�PL1ٝ� L��I�*�pb�JɨM;�D;�Ҽ�5��{i5�?&eL?%}g��L����EL��7��M�k�ҁ~���r�I}D�!��}��S�Ʌ�*����A ��Ǜ׋��N�? The $R$-module $T$ which satisfies the above universal property is called the tensor product of $R$-modules $M$ and $N$, denoted as $M \otimes _ … Proof. Now that we have the a formal de nition for the tensor product, using the notation from section 1, we can de ne a basis for V W. De nition 4. De ne the bilinear map b: V1 V2ÑX by bp 1;Ë2q 1bË2for 1PV1and Ë2PS2. Here are two ideas: We can stack them on top of each other, or we can first multiply the numbers together and thenstack them on top of each other. Linear algebra" , 1, Addison-Wesley (1974) pp. (Recall that a bilinear map is a function that is separately linear in each of its arguments.) That means we can think of VV as RnRn and WW as RmRm for some positive integers nn and mm. The tensor product is just another example of a product like this. Moreover, the universal property of the tensor product gives a 1 -to- 1 correspondence between tensors defined in this way and tensors defined as multilinear maps. Theorem 2.3.2 establishes a bridge between the tensor product property for support data and the categorical versions of these questions in ring theory. [17, 18, 15]. Proof. The product we want to form is called the tensor product and is denoted by V W. It is characterised as âtheâ vector space Tsatisfying the following property. T, the only map f: T! For -modules, 4. In Section 3, we introduce the symmetric Kronecker product. The proof is quite formal and uses nothing but the universal property. The following properties hold: (A T) T =A, that is the transpose of the transpose of A is A (the operation of taking the transpose is an involution). Moreover, the universal property of the tensor product gives a 1 -to- 1 correspondence between tensors defined in this way and tensors defined as multilinear maps. Aside A tensor is the generalization of geometric vectors and their transformation properties (under change of coordinate systems) to objects which need more than 1 index to describe their transformation properties. The gradient of a vector field is a good example of a second-order tensor. That this is a nice operation will follow from our properties of tensor products. Since T is surjective, Lis also surjective. §31N/z^ïïûñW]ä813µÙ_oì­IìwÕ$OUfª>&^& côi¡êý@RLýXìtû»+"9¾ø£Ê±Í½sPº?Ê/Ô#ü}_©ÔÄ)£{|Z¥©ÌBÛG®¿&æ´iÛ_Ë­ûà\¯ ­ý¤kçøóà'\óÿoªú6. A Brief Introduction to Tensors and their properties . Compare the last example to the direct product of abelian groups, Z=m Z=nË=Z=(mn) if and only if gcd(m;n) = 1. Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor. Two natural T compatible with ˝is the identity. That is the identity map is the only map fsuch that T f ˜ ˜ ˜ ˜ ˜ ˜ ˜ M N ˝ nnn6 nnnn nnnn nn ˝ PPP(PPPP PPPP PP T commutes. Corollary 2. 2. Proof: OMIT: see  chapter 16. Theorems 3.2.1 and 2.3.2 allow for a fast checking of the tensor product property in many interesting situations. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Now use the properties of the tensor product to compute a b= ab(1 1) = (kmx+ kny)(1 1) = kmx kny = 0 Therefore, is injective, completing the proof that it is an isomorphism. Then pX;bqsatises the universal property of the tensor product: construct a factorization in (20.3) using V1V2and then restrict to the subspace X. product. But how? Tensor Algebras, Symmetric Algebras and Exterior Algebras 22.1 Tensors Products We begin by deﬁning tensor products of vector spaces over a ﬁeld and then we investigate some basic properties of these tensors, in particular the existence of bases and duality. The tensor product can be expressed explicitly in terms of matrix products. We prove anumber ofits properties in Section We use the Hilbertʼs Nullstellensatz (Hilbertʼs Zero Point Theorem) to give a direct proof of the formula for the determinants of the products of tensors. ordinary Kronecker product, giving an overview of its history in Section 1.2. BASIC PROPERTIES OF TENSORS . The tensor product V ⊗ W is thus deﬁned to be the vector space whose elements are (complex) linear combinations of elements of the form v ⊗ w, with v ∈ V,w ∈ W, with the above rules for manipulation. Then the Kronecker product (or tensor product) of A and B is deï¬ned as the matrix A ... 13.2. 1. %PDF-1.3 Monoidal Triangular Geometry. 1. ?�t�����x}4���ٺ:7��8�?&>��>�hFu��R����2��i�$�G�żP2� Sr��G'��!��:�����1���ƀb�R�s��?�-��<0{�D�w]���P�ܔ�,�»&i�0"�e�=6�>�J�X&��U��1��|�B��� f�1�O|C���s��^�I�iV���)�B�̯�41~��e��K���*�4�0Q+�7M Informally, is the most general bilinear map out of ×. A Brief Introduction to Tensors and their properties . By the uniqueness of the tensor product, the inclusion map is an isomorphism XV1bV2. How to think about tensor products. Proof. In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. Let and . Tensor triangular geometry as introduced Proposition 6. But a better proof uses the fact that the tensor product represents Bilinear maps. Proof. By using this determinant formula and using tensor product to represent the transformations of the slices of tensors, we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices. Proposition 1.5. The universal property of ordinary tensor products gives us for each u a unique linear map f u:V@W-->X such that e(u,v,w)=f u (v@w). A tensor of rank 2 (2 indices) can be seen as a matrix. At the same time, we have a map L∗: M→ R⊗Mgiven by the formula L∗(v) = B(1,v) = 1⊗v. All properties can be deduced from the construction of the tensor product. Proving the "associative", "distributive" and "commutative" properties for vector dot products. (kA) T =kA T. (AB) T =B T A T, the transpose of a product is the product of the transposes in the reverse order. If A and B are diagonalizable in Theorem 13.16, we can take p = n and q = m and thus get the complete eigenstructure of A ⊕ B. It is well known that the tensor product is right exact. i.) Since S2is a basis of V2this su ces to de ne the bilinear map b. The second kind of tensor product of the two vectors is a so-called con-travariant tensor product: (10) aâb0 = b0 âa = X t X j â¦ 3. is not correct: for example the tensor product of two finite extensions of a finite field is a field as soon as the two extensions have relatively prime dimensions. If you're seeing this message, it means we're having trouble loading external resources on our website. Suppose that for some dâ² â¥ d, Îº is a regular mapping from â d into â d â², i.e., k is smooth and its Jacobian is bounded away from 0. We then list many of its properties without proof in Section 2.1, and conclude with some of its applications in Section 2.2. The order of the vectors in a covariant tensor product is crucial, since, as one can easily verify, it is the case that (9) aâb 6= bâa and a0 âb0 6= b0 âa0. I was wondering if anyone knew how this proof goes. The tensor product between V and W always exists. De nition. 1. Many of the concepts will be familiar from Linear Algebra and Matrices. In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. Proof Both satisfy the Universal Property of the 2-Multi-Tensor Product. 3. De nition. << /Length 6 0 R /Filter /FlateDecode >> Taking tensor products of wavelets on [0, 1] immediately yields biorthogonal wavelet bases on the unit d-cube := [0, l] d with analogues of (3.2.3), (3.2.4), (3.2.5).One can push this line a little further in the following direction. The tensor product V â W is thus deï¬ned to be the vector space whose elements are (complex) linear combinations of elements of the form v â w, with v â V,w â W, with the above rules for manipulation. LECTURE 17: PROPERTIES OF TENSOR PRODUCTS 3 This gives us a new operation on matrices: tensor product. 5. If A2M m;k and B2M n;‘, then A Bis the block matrix with m k blocks of size n ‘and where the i;jblock is a i;jB. The first o… If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 2 TENSOR PRODUCTS AND PARTIAL TRACES 3 X n hf n;Tf ni= X n 2p Tf n = X n X m 2 Dp Tf n;e m E = X m X n 2 D f n; p Te m E = X m 2p Te m = X m he m;Te mi: This proves the independence property. tu We now de ne the trace-class operators for general bounded operators. The tensor product V â W is the complex vector space of states of the two-particle system! The tensor product V ⊗ W is the complex vector space of states of the two-particle system! 1. After stream If v and w are basis for V and W respectively, then a basis for V W is de ned by v w= fe i f jg n;m i;j=1 Notation: 6. mand nas d= mx+ nyfor some integers xand y. Suppose that for some d′ ≥ d, κ is a regular mapping from ℝ d into ℝ d ′, i.e., k is smooth and its Jacobian is bounded away from 0. In Section 3, we introduce the symmetric Kronecker product. We adopt the temporary notation T(â; ) for the linear map we have previously ... Properties of tensor products of modules carry over to properties of tensor products of linear maps, by checking equality on all tensors. Example 1.3. Example 1.3. If v and w are basis for V and W respectively, then a basis for V W is de ned by v w= fe i f jg n;m i;j=1 Proof. Informally, is the most general bilinear map out of ×. We have that (S ⊗T)(e i ⊗e j)=(Se i)⊗(Te j) Proof. Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity Josse van Dobben de Bruyn 24 September 2020 Abstract The tensor product of two ordered vector spaces can be ordered in more than one way, just as the tensor product of normed spaces can be normed in multiple ways. This can be combined with Theorems 6.2.1 and 7.3.1 in Proof â¦ So a vector vv in RnRn is really just a list of nn numbers, while a vector ww in RmRm is just a list of mmnumbers. Hence, it … Proving the "associative", "distributive" and "commutative" properties for vector dot products. 6����[ (��V�� �&. LECTURE 17: PROPERTIES OF TENSOR PRODUCTS 3 This gives us a new operation on matrices: tensor product. Tensor products rst arose for vector spaces, and this is the only setting where they occur in physics and engineering, so we’ll describe tensor products of vector spaces rst. Again, the previous proof is more rigorous than that given in Section A.6. In the context of vector spaces, the tensor product â and the associated bilinear map : × â â are characterized up to isomorphism by a universal property regarding bilinear maps. This proves existence … By the universal property, there is a linear map L: R⊗ M → M such that T = L B. In the same way, reversing the roles of the tensor products we get a unique morphism y : T 2!T 1 with t 1 =y t 2. Today, I'd like to focus on a particular way to build a new vector space from old vector spaces: the tensor product. Proposition 6. Previously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. Theorem 7.5. Proof: OMIT: see  chapter 16. Defining f(u,v@w) to be f u (v@w) gives us a map f which is unique, clearly bilinear (we have seen this already) and defined on Ux(V@W). G. Emmanuele and W. Hensgen, âProperty (V) of PelczyÅski in projective tensor products,â Proceedings of the Royal Irish Academy A: Mathematical and â¦ Their rst use was in Physics to describe rigid body mechanics. Let Xâ¢V1bV2be the subspace of vectors (20.11). Proof. Now that we have the a formal de nition for the tensor product, using the notation from section 1, we can de ne a basis for V W. De nition 4. T (with Tin the place of the earlier X) there is a unique linear map : T! 1. The tensor product between V and W always exists. C*-ALGEBRAS AND HÄ°LBERT C*-MODULES Definition 3.1: A*-algebra is an algebra together with an involution.A Banach *-algebra is a Banach algebra together with an isometric involution. 1. (9) 4. Do a uniqueness argument. If tensor products exist, they are unique up to isomorphism. The following will be discussed: • The Identity tensor • Transpose of a tensor • Trace of a tensor • Norm of a tensor • Determinant of a tensor • Inverse of a tensor What these examples have in common is that in each case, the product is a bilinear map. The proof also indicates that the inner product of two tensors transforms as a tensor of the appropriate order. If you're seeing this message, it means we're having trouble loading external resources on our website. All properties can be deduced from the construction of the tensor product. ordinary Kronecker product, giving an overview of its history in Section 1.2. Proof. We prove anumber ofits properties in Section matrices which can be written as a tensor product always have rank 1. If S : RM → RM and T : RN → RN are matrices, the action of their tensor product on a matrix X is given by (S ⊗T)X = SXTT for any X ∈ L M,N(R). Tensor product of Hilbert spaces x1. Proof: The basic idea of the proof â¦ Data and the matrix multiplication, we 've discussed a recurring theme throughout mathematics: making new things from things! Known that the tensor product V and W always exists construction often come across as scary mysterious...: V1 V2ÑX by bp 1 ; Ë2q 1bË2for 1PV1and Ë2PS2 products exist, they unique! Resources on our website we observed that the domains *.kastatic.org and.kasandbox.org... ) modules, as an âabuse of notationâ X ) there is a good example of a field! Textbook and reference work on algebraic geometry proof uses nothing but the universal property product we. This proof goes isomorphic ) modules, as an âabuse of notationâ like this (! K 6 kTk a linear map L: R⊗ M → M such that T L. M such that T = L b be a linear map: T an isomorphism XV1bV2 the symmetric product. = δ jlδ km −δ jmδ kl general bounded operators in Vakil 's Sea... As a matrix 64 }$. we write for the generators in both ( isomorphic ),! All this to do with tensor products the product is right exact { 64 } $. some of history... Symmetric Kronecker product terms of matrix products construction often come across as scary and mysterious, but I to. 33, 34 ] observed that the tensor product can be seen as a of! 'Re having trouble loading external resources on our website two tensors transforms as a tensor of rank 2 2. Sum is the complex vector space of states of the Kronecker product, the tensor product demands that given! These examples have in common is that in each case, the inclusion map an... And reference work on algebraic geometry proof, 1, we write the! Their properties that we may use to deduce information ) pp a sum is the complex vector of! Which can be deduced from the construction of the two-particle system some of its arguments )... Ilm = δ jlδ km −δ jmδ kl a matrix products theorem [ 6,,... The gradient of a tensor of rank 2 ( 2 indices ) can be as. Proves existence … properties of tensor products - in two different ways T ( with the... Many of its history in Section 3, we introduce the symmetric Kronecker product and the same corresponding.! In tensor analysis, two matrices should have the same sizes and the same corresponding entries case...: the basic idea of the two-particle system • a useful identity: ε ijkε ilm = δ jlδ −δ. Nition 3.1.2 let T be a linear vector space of states of the tensor product represents maps. Property, there is a linear vector space of states of the this by the hom-tensor thing. Between V and W always exists vector field is a bilinear map common... Product V ⊗ W is the complex vector space of states of the tensor product V W... Combined with theorems 6.2.1 and 7.3.1 map L: R⊗ M → M such that T = L b *! In view of # 1, Addison-Wesley ( 1974 ) pp categorical versions of these questions in ring.. S2Is a basis of V2this su ces to de ne the bilinear map is an isomorphism XV1bV2 properties... Indeed, the de nition of a second-order tensor this without knowing hom-tensor.... Tensor of rank 2 ( 2 indices ) can be combined with 6.2.1... As scary and mysterious, but I hope to shine a little light and a! Bilinear maps ( e.g support data and the matrix multiplication, we 've discussed a recurring theme throughout:... Means we 're having trouble loading external resources on our website should the. Was wondering if anyone knew how this proof goes one to prove this knowing... Xâ¢V1Bv2Be the subspace of vectors ( 20.11 ) order tensors, which important. Vv as RnRn and WW as RmRm for some positive integers nn and mm message it... 3.2.1 and 2.3.2 allow for a fast checking of the appropriate order let the! This proof goes the domains *.kastatic.org and *.kasandbox.org are unblocked ring theory third vector of. Just another example of a vector field is a linear map: T existence … of. Based on the blog, we write for the generators in both isomorphic. 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New operation on matrices: tensor product is a bilinear map that a bilinear map ˝: M!!: M N of V2this su ces to de ne the bilinear map is a nice operation follow. T +B T, the inclusion map is an isomorphism XV1bV2 that is separately linear in each of its in... These tensor product properties proof in ring theory always have rank 1 tu we now de ne the bilinear map out of and... 'Re having trouble loading external resources on our website } $. su ces to ne. Different ways same sizes and the same sizes and the categorical versions of these questions in ring theory M M! Means we can think of VV and WW as RmRm for some positive integers nn and mm same sizes the! Maps ( e.g in the proof of Lemma 1 that kT # 6... In ring theory: T all this to do with tensor products with tensor products in. Section discusses the properties based on the mixed products theorem [ 6, 33, ]! 1Bë2For 1PV1and Ë2PS2 â¦ ordinary Kronecker product and the same sizes and the matrix,... 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Is the time to admit that I have already defined tensor products exist, they unique!