Question

Consider the mapping from $V \otimes V^$ to $\operatorname{Lin}(V ; V)$ which sends $v \otimes f\left(v\right.$ in $V, f$ in $\left.V^\right)$ to the linear mapping from $V$ to $V$ which sends $v^{\prime}$ in $V$ to $\left[f\left(v^{\prime}\right)\right] v$. Prove that this is a linear mapping and that it is a monomorphism. Prove that it is an isomorphism if and only if $V$ is finite-dimensional.

   Consider the mapping from $V \otimes V^*$ to $\operatorname{Lin}(V ; V)$ which sends $v \otimes f\left(v\right.$ in $V, f$ in $\left.V^*\right)$ to the linear mapping from $V$ to $V$ which sends $v^{\prime}$ in $V$ to $\left[f\left(v^{\prime}\right)\right] v$. Prove that this is a linear mapping and that it is a monomorphism. Prove that it is an isomorphism if and only if $V$ is finite-dimensional.

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Mathematical physics

Mathematical physics

Robert Geroch 1st Edition

Chapter 14, Problem 99

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Step 1

Consider the mapping \( \Phi: V \otimes V^* \to \operatorname{Lin}(V; V) \) defined by \( \Phi(v \otimes f)(v') = f(v')v \) for \( v, v' \in V \) and \( f \in V^* \). Show more…

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Consider the mapping from $V \otimes V^*$ to $\operatorname{Lin}(V ; V)$ which sends $v \otimes f\left(v\right.$ in $V, f$ in $\left.V^*\right)$ to the linear mapping from $V$ to $V$ which sends $v^{\prime}$ in $V$ to $\left[f\left(v^{\prime}\right)\right] v$. Prove that this is a linear mapping and that it is a monomorphism. Prove that it is an isomorphism if and only if $V$ is finite-dimensional.

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