Isomorphism between tensor product of dual space and space of bilinear maps

Asked Jan 24 '15 at 06:58

Active Jan 24 '15 at 06:58

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I am trying to show $V^* \otimes W^* \simeq L_2(V \times W,\mathbb{R})$ using the definition here.

Hence I try to show $\phi:V^* \otimes W^* \to L_2(V \times W,\mathbb{R})$ defined by $\phi(\alpha\otimes \beta)=\Phi_{\alpha,\beta}$, where $\Phi_{\alpha,\beta}(v,w)=\alpha(v)\beta(w)$ is an isomorphism. If I assume $\phi$ is a linear map, I can show it's bijective. But I don't know how to show the linearity of $\phi$, in particular, I don't know how the addition and scalar multiplication in tensor product space are defined.

If $\alpha\otimes\beta,\gamma\otimes\delta\in V^* \otimes W^*$, what is $\alpha\otimes\beta+\gamma\otimes\delta$? and what is $\phi(\alpha\otimes\beta+\gamma\otimes\delta)$ in $L_2(V \times W,\mathbb{R})$?

edited Apr 13 '17 at 12:20

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asked Jan 24 '15 at 06:58

John

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What is your definition of the tensor product of two vector spaces? – Travis Willse Jan 24 '15 at 07:25
Use the universal property. (This is the definition of the tensor product.) – Martin Brandenburg Jan 24 '15 at 07:25
@Travis I am using the definition here. It is cited from Gallot Riemannian Geometry book. – John Jan 24 '15 at 07:29
@MartinBrandenburg I don't quite understand how the universal property is related with the linearity of $\phi$. May you explain more? – John Jan 24 '15 at 07:32

Isomorphism between tensor product of dual space and space of bilinear maps

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