Tensor products allow us to build "linear" objects from "multilinear" ones. It can refer to: basic ones from linear algebra/module theory, or more sophisticated versions from differential/algebraic geometry (bundles/sheaves), functional analysis (Hilbert/Banach/locally convex spaces), or in their ...

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sections of tensor product of sheaves of modules

I am confused about notation concerning tensor products of sheaves of modules. I know that given a ringed space $X$ and $\mathcal{O}_X$-Modules $\mathcal{F}$ and $\mathcal{G}$ their tensor product is ...
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35 views

Definition of connection on vector bundle

A connection on a vector bundle $E$ is a map $ D:\Gamma(E)\rightarrow \Gamma(T^*(M)\otimes E)$ satisfying 1) For any $s_1,s_2\in \Gamma(E)$, $D(s_1+s_2)=Ds_1+Ds_2$ 2) For $s\in \Gamma(E)$ and ...
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30 views

What is this map between tensor spaces called? (Change of coordinates).

Let $V$ and $W$ be finite dimensional vector spaces. Let $A: V \to W$ be linear. Define the map $A^* : (W^*)^{\otimes r} \to (V^*)^{\otimes r}$ by $(A^*\alpha)(w_1, w_2, \ldots, w_r) = \alpha(Aw_1, ...
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What is an operational definition for a tensor?

The two tensor definitions I'm familiar with, by transformation rules, and as a map from a tensor product space to the reals, don't tell me what a tensor does, and TTBOMK (to the best of my knowledge) ...
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Natural Isomorphism between $V^*\otimes W^*$ and $\mathcal L^2(V,W; F)$.

I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and ...
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Basic Question Regarding the Universal Property of the Tensor Product.

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space ...
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Natural Transformaton $\text{Hom}(V,W)$ and $W\otimes V$

Something of this form has already been answered here: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? I'm starting introductory category theory stuff, and I'm looking for some help. I ...
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Derivation of tensor (with einstein summation)

Let $f$ be a scalar defined by: $$f = a_{iik}a_{kjj} (1 \leq i, j, k \leq 3)$$ in which $a$ is a third-rank tensor $a_{ijk} = a_{jik}$ and the summation convention for repeated indices is employed. ...
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52 views

Injective tensor product

We know that $c_0\check\otimes c_0=c_0(c_0)$ where $\check\otimes$ is the injective tensor product. is the following still true? $$c_0\check\otimes l^\infty = l^\infty(c_0).$$ Thank you for your help ...
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36 views

Showing $S^{-1}(M \otimes_{A} N) \cong S^{1}M \otimes_{S^{-1}A} S^{-1}N$

One of the propositions in Atiyah-MacDonald's Commutative Algebra states $$S^{-1}(M \otimes_{A} N) \cong S^{-1}M \otimes_{S^{-1}A} S^{-1}N.$$ The proof in the text states that one should use that ...
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90 views

Tensor product notation for $(x,y)\mapsto f(x)g(y)$

I often stumbled across (variations of) the notation $f\otimes g$ for $(x,y)\mapsto f(x)g(x)$. If $f\in V^*$, $g\in W^*$ for vector spaces $V,W$ over ${\bf K}$ then $f\otimes g:x\otimes y\mapsto ...
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92 views

Is a product of two Noetherian schemes over Spec $\mathbb Z$ a Noetherian scheme?

In Hartshorne's proof of Proposition 6.6 in Chapter 2, he says that if $X$ being Noetherian implies $X\times\mathbf A^1$ is "clearly" Noetherian. I assume this is because $X$ can be covered by affine ...
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1answer
17 views

Module tensor product of an element with zero

Is it true that for any two R-modules, N, M, that the tensor product $n \otimes 0 = 0 \otimes m = 0 \space\space \forall \space n \in N, m \in M$? The reason I am thinking this is true is because ...
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1answer
26 views

Isomorphism between different definitions of symmetric tensors

Hi I was working through symmetric tensors (technically in a differential manifold, but we defined it abstractly). We defined them first in the usual way: Consider $A$ an integral domain and ...
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18 views

I have problem in understanding the relation?

If we have a relation like this: $$\frac { \partial }{ \partial x_\beta } (\varepsilon_{ij\alpha} x_j T_{\alpha \beta}) =\left[ \nabla \cdot(\vec x \times \overset {\leftrightarrow}{T} ) \right]_i$$ ...
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2answers
51 views

To show $L\otimes_K \text {End}_K(V)\cong \text{End}_L(L\otimes_K V)$

Let $L/K$ be field extension and let $V$ be a $K$-vector space. Then do we have an isomorphism $$L\otimes_K \text {End}_K(V)\cong \text{End}_L(L\otimes_K V)$$ as $L$-algebras? My attempt: for ...
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53 views

Localization of tensor products

Assume $A$ and $B$ are finite type $k$-algebras over a field. Consider the tensor product $C = A\otimes_k B$. There are maps $A\to C$ and $B\to C$. Given a prime ideal $p \subseteq C$, I can ...
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45 views

Does covariant derivative commute with “generalized contraction”_ About the proof of 2nd Bianchi identity

I am reading the proof of second Bianchi identity on wiki. In the proof, it says the following condition must satisfy: $$((D_X R) (Y,Z)) (W) + R (D_XY,Z) W + R(Y,D_XZ) W + R(Y,Z) D_X W = D_X ...
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1answer
39 views

$v \times w$ is a bilinear map, antisymmetic and $u \times w =0 \Leftrightarrow $ collinear in tensor product

This is my Attempt for part (b): Let's define: $$\Phi: \mathbb{R}^2 \times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 \otimes \mathbb{R}^2 $$ by the following action: $$\Phi(v \times w) = v \otimes ...
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1answer
54 views

Basic property of a tensor product

I think that this might follow from a basic properties of tensor products, but I am q bit stuck... Let $A$ be a $k$-algebra. Let $l/k$ be a finite field ext. of $k$. Suppose $A \otimes_k l$ is an ...
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1answer
44 views

How to find a non-surjective and non-injective tensor products of the formal completion?

Let $A$ be a commutative ring with unit endowed with $I$-adic topology where $I$ is the ideal of $A$. Let $\hat A$ be the formal completion of $A$ for the $I$-adic topology, and $M$ an $A$-module. Let ...
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Integration over a second order tensor

I would like to compute the mean value of a second order tensor $\mathbf{T}$ expressed in planar cylindrical coordinates. The mean value for any second order tensor is (reference [1] page 101) ...
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How to compute the inertial tensor ${\bf{J}} _{\Omega}$?

Suppose that $\Omega$ is a body of revolution of the function $y=f(x), a\le x \le b$ around the $x$-axis, where $f(x)>0$ is continuous. How to compute the inertial tensor ${\bf{J}} _{\Omega}$? ...
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1answer
38 views

How to prove some formulations aboult Kronecker product?

The Kronecker product has some properties as the wikipedia http://en.wikipedia.org/wiki/Kronecker_product. For the sake of simplicity, we denote $\mathbf{U}_{M}^T\otimes ...
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1answer
25 views

Tensor algebra of $Μ$ over $R$

Let $R$ be a commutative ring and $Μ$ an $R$-module. Consider the $R$-module $T_{R}(M)={\displaystyle \oplus_{n=0}^{\infty}}M^{\otimes n}$ and show that it can be endowed with the structure of an ...
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27 views

Spatial tensor product of operator spaces

If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into ...
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1answer
25 views

Tensor Product: Boundedness

This thread is just a note. Given Hilbert spaces. Then boundedness will be inherited: $$A,B\text{ bounded}\implies A\otimes B\text{ bounded}$$ Especially, the bounds multiply: $$\|A\otimes ...
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1answer
28 views

Tensor Product: Denseness

This is thread is just a note. Given Hilbert spaces. Then denseness will be inherited on tensor products: $$\mathcal{D},\mathcal{E}\text{ dense}\implies\mathcal{D}\otimes\mathcal{E}\text{ dense}$$ ...
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1answer
25 views

Tensor Product: Closability

This was a real question of mine. Given Hilbert spaces. Then closability will be inherited on tensor products: $$A,B\text{ closable}\implies A\otimes B\text{ closable}$$ For simple tensors this is ...
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1answer
64 views

What is ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$?

How exactly can I prove that ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$ equals the linear space $M$ spanned by 'simple fucntions' $\phi\in{\cal D}(\Omega_0\times\Omega_1)$ of the form ...
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1answer
31 views

Tensor Product: Identification

This is meant as note. Given a measure space and a Hilbert space. Then there's an identification: $$\mathcal{L}^2(\mu)\hat{\otimes}\mathcal{H}\cong\mathcal{L}^2_\mathcal{H}(\mu):\quad ...
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1answer
27 views

Tensor Product: ONB

This thread is just a note. Given Hilbert spaces. Consider their hilbertian tensor product: ...
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33 views

Can someone show why this tensor product of matrices is correct?

I am trying to understand tensor products better. I decided to find an example and try to understand it. I read somewhere the the following is a correct application of the tensor product for these ...
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1answer
61 views

Evaluating contractions of a tensor product

Consider $T = \delta \otimes \gamma$ where $\delta$ is the $(1,1)$ Kronecker delta tensor and $\gamma \in T_p^*(M)$, the co-tangent space over some manifold $M$. Evaluate all possible contractions of ...
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2answers
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What is the module structure here?

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$, and let $M$ be an $A$-module. I want to turn the following object into an $A/\mathfrak{m}$-module: $$A/\mathfrak{m} \otimes_A M$$ I ...
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2answers
164 views

What does it mean to tensor with $\mathbb{Q}$?

At our algebraic geometry seminar I often hear that something is 'tensored with $\mathbb{Q}$', e.g. a ring of endomorphisms. This phrase seems to have some intuitive meaning that I don't know. What ...
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256 views

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$?

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$? Here tensor product is over the ring $\mathbb Z$ and $\mathbb Z[[X]] $ denotes formal power series over $\mathbb Z$. I think ...
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1answer
58 views

Canonical map is injective

Let $A$ and $B$ be commutative rings, and let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that for any $A$-module $M$, the canonical map $M\to M\otimes_AB$ is injective? I was ...
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0answers
44 views

How to prove the formulation of mode-$n$ matricization and preclusive mode-$n$ product?

The mode-$n$ product of a tensor $\mathcal{X}=[x_{i_1,\ldots,i_M}]\in \mathbb{R}^{I_1\times \cdots \times I_M}$ and a matrix $\mathbf{U}=[u_{i_m,j}]\in \mathbb{R}^{I_m\times J}$ is denoted by ...
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Inclusion $\mathbb{Q} \hookrightarrow \mathbb{R}$ induces an injection in scalar extensions

Question Given a torsion-free $\mathbb{Z}$-module (aka. abelian group) $G$, let $i: \mathbb{Q} \hookrightarrow \mathbb{R}$ be the inclusion. I want to show that $$ i \otimes \mathrm{id}: \mathbb{Q} ...
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Tensor of tensored categories

Given two $V$-categories $C$ and $D$ tensored over a symmetric monoidal category $V$, could I form the "tensor" of $C$ and $D$? More precisely, is there a $V$-category $T(C,D)$ such that $V$-functors ...
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1answer
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Confusion in the tensor product homomorphism in representations

Let $k$ be a field and $G$ a group. Let $V$ and $W$ be two representations. And $V \otimes _k W$ be their tensor product which itself a representation with $G$ act on $V \otimes _k W$ by $g ( v ...
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1answer
43 views

Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
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1answer
40 views

Show that the trace of the operator $S \wedge T$ is zero

I have some difficulties with the following problem: Let $V$ be a finite dimensional vector space over $\mathbb{K}$. Let $S,T \in L(V,V)$. Show that the trace of the operator $S \wedge ...
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65 views

Question about tensor products, decomposable tensors, …

I need some help with the following problem: Let $V_1,\ldots,V_m$ be finite dimensional vector spaces over $\mathbb{K}$. Let $\varphi \in L(V_1,\ldots,V_m;U)$ such that $Im(\varphi)=U$. ...
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1answer
37 views

What is the completed projective tensor product of compactly supported smooth functions on two manifolds?

Let $M$ and $N$ be smooth manifolds (not necessarily closed). It is a lovely fact that $$C^\infty(M \times N) \cong C^\infty(M) \hat{\otimes}_\pi C^\infty(N).$$ See, for the instance, the book ...
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1answer
47 views

Computation rules for tensor products and inner products

I'm studying distributions and just came along the formula $$\langle f\otimes g,\phi\otimes\psi\rangle=\langle f,\phi\rangle\langle g,\psi\rangle.$$ I understand what that means in the context of ...
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tensor products of $\mathbb Z$-modules over quotient rings

Please I have a question concerning Tensor products. What is $\mathbb Z/a \mathbb Z$ tensor $\mathbb Z /b \mathbb Z$ over $\mathbb Z/c \mathbb Z$ where $a$,$b$ and $c$ are positive integers? Maybe a ...
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Dimension of sum of permutations of tensor products of vector spaces

Sorry for the mouthful of a title! Suppose I have two finite vector spaces $W,V$ with bases $\{w_1\dots w_p\}$ and $\{v_1\dots v_q\}$. Consider some subspace $S$ of $W\otimes V$ of dimension $m$ ...
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Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...