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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Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
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41 views

Why is this a bounded operator?

Let $\mathcal{H}$ be the Hilbert space $l^2(\mathbb{N})\otimes l^2(\mathbb{Z})$. I want to prove that the operator $T$ defined by $$T:=\sum_{k=1}^{\infty}{\sqrt{1-q^{2k}}e_{k-1,k}\otimes 1}$$ is a ...
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How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
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22 views

Eigenvalues of integrals over similar matrices

Let $\rho = \rho(x)$ be a $2\times2$ matrix (don't know if it is necessary, but $\rho$ is a density operator) and $I$ be the (two-dimensional) identity matrix. I have two matrices $A$ and $B$, where ...
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Is $(x)\otimes_{k[x]/(x^2)}(x)$ zero?

I am trying to decide if $(x)\otimes_{k[x]/(x^2)}(x)$ is zero. So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live in ...
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tensor products of Hopf algebras

Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ ...
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37 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
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1answer
23 views

Copower functor

Computing copowers and "tensoring with sets" often means the same thing. If a locally small category $\mathcal{C}$ has coproducts and if $S$ is a set then for any object $C\in\mathcal{C}$ the copower ...
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40 views

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...
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41 views

What is $\mathrm{dim}(\mathrm{Sym}(\mathrm{Herm}(H)^{\otimes N})$?

The totally symmetric subspace of $(H^k)^{\otimes N}$, with $H^K$ a $k$-dimensional Hilbert space, has dimension $\binom{N+k-1}{k-1}$. But I now want to know the dimension of the totally symmetric ...
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42 views

Tensor product isomorphic to a free module

Is it true that if $R$ is a domain with quotient field $K$ and $M$ is a finitely generated torsion-free $R$-module then $M\otimes_R K$ is isomorphic with $K^n$ for some $n$? I know that the first is ...
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33 views

A question about 1.0.3 in Grothendieck's EGA

In (1.0.3), Grothendieck states that, given non-commutative rings $A$ and $B$, a homomorphism $\varphi : A \to B$, and a left ideal $\mathfrak{J}$ of $A$, the left ideal $B\mathfrak{J}$ of $B$ ...
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Solving a quadratic vector/tensor equation arising from connected Markov chains

I have a discrete-time finite-state aperiodic irreducible Markov chain, which is composed of $m$ identical component sub-chains. With probability $1-\mu$, in each time step each of these chains ...
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45 views

Basic Confusion About Tensor Products

Let $A$ and $B$ be subspaces of vector spaces $V$ and $W$ respectively. Given $a\in A$ and $b\in B$, there are two possible interpretations of $a\otimes b$: we can think of it as a member of ...
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Doubt regarding the definition of tensor product

I refer to the following image from "Atiyah-Macdonald", which describes the tensor product of modules. I don't understand how D forms a module. How exactly should I had two elements in D?
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29 views

Some calculations with skew forms and wedge product

I have some problems with the language of multilinear forms. I have to prove that if $dim(V)\le 3$, then every $\omega\in\Lambda^q(V^\ast)$ is such that $\omega\wedge\omega=0$. I consider the case ...
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37 views

Basis of a vector space using basis of tensor product and universal property

I already proved that if $\{u_i\}$ is a basis for $U$, $\{v_i\}$ basis for $V$, then $\{u_i \otimes v_j\}$ is a basis for $U \otimes V$. Now, I am trying to prove that if $g:U \times V \to W$ is ...
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27 views

Example tensor representation

Studying from Roman's Advanced Linear Algebra I got stuck with the following exercie, I want to find two vector spaces $U$ and $V$ and a nonzero vector $x\in U \otimes V$ that has at least two ...
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39 views

Example of a bilinear map whose image is not a subspace

I am looking for an example of a bilinear map $\tau:V \times V \to W$ whose image $im(\tau)=(\tau(u,v):u,v \in V)$ is not a subspace of $W$. I considered the tensor map $\tau:U \times V \to U \otimes ...
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43 views

Isomorphism between dual space and bilinear forms

Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ ...
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28 views

How to compute tensor product between two functions

Let $f$ and $g$ are two functions defined on $[0,1]$ and taking values in a Hilbert space $H$. Then how to define and compute the tensor product between $f$ and $g$, namely $f\otimes g$? I know how ...
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58 views

Star operator in the simplest form

Let $E$ together with $g$ be a inner product space(over field $\mathbb R$) , $\text{dim}E=n<\infty$ and $\{e_1,\cdots,e_n\}$ is orthonormal basis of $E$ that $\{e^1,\cdots,e^n\}$ is its dual ...
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Universality of Tensor product as defined here.

I'm defining Tensor Product as Berberian like $V_1\otimes V_2=\mathcal{L}(V_1^*,V_2)$ like here and inductively as $V_1\otimes V_2\otimes V_3=(V_1\otimes V_2)\otimes V_3$. I have defined the map ...
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39 views

tensor-operator- schatten norm

let $H_1$ and $H_2$ are hilbert spaces with dimention $n$ and $m.$ ‎assume $$\varphi:\mathcal{B}(H_1)\hat{\otimes}\mathcal{B}(H_2)\to\mathcal{B}(H_1\hat{\otimes}{H_2)},$$ which is defind as ...
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23 views

Norm of $n \times n \times n$ Tensor

Given a real positive semi definite matrix, $A\in R^{n\times n}$ and a real matrix $F \in R^{b\times n }$, we have the following inequality: $\|FAF^T\| \le \|FF^T\| \|A\|$. However, I am wondering ...
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42 views

Equivalence between definitions of Tensor Products (Of Vector Spaces)

I've read the definitions in the book of Kostrikin and in the book of Sterling Berberian. The book of Berberian gives a definition for the product of two spacees, Kostrikin gives it more generally. ...
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28 views

Tensor Products of bimodules over commutative rings

Suppose that $ R $ and $ S $ are commutative rings with identity, $ R \subset S $, $ 1 _{R} = 1_{S} $, $ M $ is a $ (S,R)$-bimodule, $ N $ is a $ (R, S)$-bimodule, $ T = M ...
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Definition of tensor product using pushout.

Let $M, N$ be right and left $A$-modules respectively. Then we have actions $$ \varphi_M: M \times A \to M, \\ \varphi_N: A \times N \to N. $$ There is a definition of $M \otimes_A N$ as follows. We ...
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40 views

When is the tensor product commutative?

I am working with the the tensor product $-\otimes_R -$ over some noncommutative ring $R$. Is the tensor product always commutative if $R$ is commutative? If so, can the tensor product be commutative ...
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87 views

Properties of tensor product of modules

Let $M'$ be a submodule of $\mathbb{Z}$-module $M$, and let $i:M'\rightarrow M$ be a natural monomorphism. How to prove the following theorem ? : $i\otimes 1_N:M'\otimes_{\mathbb{Z}} N ...
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Is it true that $M\otimes_A F\simeq M^{(I)}$?

Let $A$ be an $R$-algebra ($R$ is a commutative ring with identity $1_R$) and suppose $F$ is a left free module over $A$. Is it true that $$M\otimes_A F\simeq M^{(I)}$$ for any right module $M$ over ...
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1answer
31 views

Another approach to $A/I\otimes_A A/J\simeq A/(I+J)$?

The same question appears here $A/ I \otimes_A A/J \cong A/(I+J)$ however I'm looking for a different approach. Let $A$ be an algebra, $I$ a right ideal and $J$ a left ideal. I'd like to show ...
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1answer
68 views

When is $M\otimes N$ a module?

So suppose we have a $R$-$S$-bimodule $M$, and an $S$-$T$ bimodule $N$, then we can construct the abelian group $M\otimes_S N$. Under what conditions could we make this a module? I would expect this ...
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Revert problem in Kronecker tensor product

I have a question related to reverting kron tensor product. As We can see the in below example: ...
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37 views

Ring homomorphism of tensor product of algebras

Let $B, C$ be two $A$-algebras, $f:A \to B, g: A\to C$ the corresponding ring homomorphisms. From this we can construct an $A$-algebra $B \otimes _A C$ and the mapping $ a \mapsto f(a) \otimes ...
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90 views

Determinant of the Transpose of an Operator.

Let $V$ be a vector space over a field $F$ of characteristic $0$. A linear operator $T$ on $V$ induces a linear operator $\Lambda^k T:\Lambda^k V\to \Lambda^k V$ such that $\Lambda^k T(v_1\wedge ...
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Canonical linear mapping is bijective

Let $V$ be a $K$-vector space with finite dimension. Proof that mapping: $V^* \otimes V \rightarrow {\rm End}_K(V), \ h\otimes a\mapsto (x\mapsto h(x)a)$ is bijective. So we have one mapping, which is ...
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1answer
61 views

Kernel of the Symmetrizing Map $Sym:\bigotimes^k V\to \bigotimes^k V$

$\DeclareMathOperator{\sym}{Sym}$ Let $V$ be a finite dimensional vector space over a field of characterisitc $0$ and $\sym:\bigotimes^k V\to \bigotimes^k V$ be the map given by $$ ...
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1answer
37 views

Tensor Product of Vectors

let $S,T$ be respectively $k$-, $n$-tensors; $k,n>0$. Then we define the tensor product $$ T \otimes S(x_1,x_2,\ldots,x_{k+n}) := T(x_1,\ldots,x_k) S(x_{k+1}, \ldots, x_{k+n}) $$ (their product as ...
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Evaluation of polynomials at tensor products

Let $S,T$ be $R$-algebras, $f \in S[X]$ a polynomial. In my notes it says you can easily lift $f$ to a polynomial $f'$ in $(S \otimes T)[X]$. But I have no idea what $f'(s \otimes t)$ is. My ...
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Example of two field extensions such that their tensor product is not a field

Example of two fields $K$ and $L$, which are extensions over $k$, such that $K\otimes_k L$ is not a field. Here is what I did. But I am a little bit unsure. Can someone suggest anything, or perhaps ...
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Rank of tensor product of morphisms

Let $R$ be a commutative, noetherian, unital ring, $F$ and $G$ two projective $R$ modules, $\phi: F\to G$ a module morphism and $M$ a finitely generated $R$ module such that $$\phi \otimes M := \phi ...
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Why not take the tensor product of two left modules in this way? [duplicate]

Let $A,B$ be two left $R$-modules. I was wondering if we then can form the tensor product of $A$ and $B$ by the free abelian group on $A \times B$ divided out by the span of the following elements ...
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1answer
33 views

Direct Sum vs. Direct Product vs. Tensor Product

There are a lot of questions like this all over the site, but I cannot find one that resolved my confusion- what are the formal definitions of direct sums, direct products, and tensor products (in the ...
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Tensor product of bounded analytic functions

Let $H^\infty(\mathbb{D})$ denote the set of functions holomorphic and bounded on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Conseqently, $H^\infty(\mathbb{D}^n)$ denotes the set of bounded ...
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Dense subsets in tensor products of Banach spaces [duplicate]

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
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55 views

Dense subsets in tensor products of Banach spaces

Assume $B_1$ and $B_2$ are Banach spaces of univariate functions. Moreover, assume that the sets $D_1 \subset B_1$ and $D_2 \subset B_2$ are dense with respect to the respective norms ...
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124 views

Isomorphism between $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$ [closed]

I hope you can help me with this: Show that $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$ are isomorphic, and specify an isomorphism. Thanks.
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About the tensor product identity: $A=B \otimes C = B \otimes I + I \otimes C$

I am reading about Chern classes in Nakahara's Geometry, Topology and Physics, and am having trouble understanding the equation $$ A=B \otimes C = B \otimes I + I \otimes C \tag{1}$$ where $A,B,C$ are ...
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1answer
51 views

Showing the tensor product spans a vector space, and interpreting things

The question is as follows: Prove that the tensor products $\tau\otimes\theta\in L(V,W;F)$ where $\tau\in V$ (suspect typo for $\tau\in V^*$ - the book even defines this above) and $\theta\in W$ ...