# Tagged Questions

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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### Vector calculus - Are these expressions equivalent?

In a book I have come across the following expression $E(\nabla \cdot E)=E\cdot(\nabla E)$, where $E=\sum_i E_i \vec{e}_i$. Unfortunately I could not prove this, when I calculate both expression ...
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### Measure space of cartesian product

If $X$ and $Y$ are two locally compact Hausdorff spaces, we have their associated measure spaces $M(X)$ and $M(Y)$. Is $M(X\times Y)$ describably in terms of the two smaller spaces? My guess is ...
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### Showing that a certain map $\operatorname{End}(V) \otimes \operatorname{End}(W) \rightarrow \operatorname{End}(V\otimes W)$ is an isomorphism

Let $V, W$ be vector spaces and $A\in\operatorname{End}(V), B\in\operatorname{End}(W)$ endomorphisms. We can define a linear map $A\tilde\otimes B \in \operatorname{End}(V\otimes W)$ by ...
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### Different ways of Proving the existence of Tensor Product

This is Just a curosity. Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($M \otimes N$,u) (where $M \otimes N$ is a $A$-module ...
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### Uniqueness of duals in a rigid monoidal category [duplicate]

Let $V$ be a simple object in a rigid monoidal category. A simple object $Y$ is the left dual of $V$ if and only if there is an epimorphism $V \otimes Y \to 1$. It is well-known that left (and ...
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### Koszul sign convention and symmetric group action on the graded n-th tensor product

Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ and $W_\bullet = (W_k)_{k \in \mathbb{Z}}$ be two graded vector spaces on 0 caracteristic field. We define the tensor product of $V_\bullet$ by $W_\bullet$ ...
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### Proving the Leibniz Rule for Lie Derivatives of tensor fields.

I am learning some Differential Geometry on my own in preparation for a course I'm starting in October, and one of the exercises in the notes I'm using is to check that the Lie Derivative satisfies ...
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### Find eigenvalues of operator

Let $A$ be a linear operator which acts on the vector space $V=\langle x_1,x_2, \ldots,x_n\rangle$. Suppose we know its eigenvalues - $\lambda_1, \lambda_2, \ldots, \lambda_n.$ Now consider the ...