Covector applied to a tensor product

Definition

Given two vector spaces, $(V,F)$ and $(W,F)$ and a covector in the dual space of $V$, $f^*\in V^*$, with:

• $$f^*:V\rightarrow F$$

We can define a map, denoted $f^*:V\otimes W\rightarrow W$ as follows:

• $$f^*:V\otimes W\rightarrow W$$ by $f^*(\sum^k_{i=1}v_i\otimes w_i)=\sum^k_{i=1}f^*(v_i)w_i$

Proof of claims

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Claim: This is a linear map

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Simple, just do $f^*(\alpha(v_1\otimes w_1)+\beta(v_2\otimes w_2))$ and find this gives the expected result

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Claim: This is well defined

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Given two representations for the same tensor products, we must show that $f^*$ of them both is well defined. See p48 LAVEP