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A multilinear map from the product $V_1\times\ldots\times V_n$ of vector spaces over the same field $K$ to another vector space $W$ over $K$ is a map $\phi$ such that if we fix vectors $$v_1\in V_1,\ldots,v_{i-1}\in V_{i-1},v_{i+1}\in V_{i+1},\ldots,v_n\in V_n,$$ then the map $\bar\phi:V_i\to W$ given by $$\bar\phi(v_i)=\phi(v_1,\ldots,v_{i-1},v_i,v_{i+1},\ldots,v_n)$$ is a linear map.

I have two questions about generalizing this.

Question 1. Why is the definition limited to finite products? Couldn't we do it just as well for infinite products?

Question 2. Can't we do it for any kind of structures that can be "producted"? For example, I think we could do it for groups by changing all vector spaces to groups in the definition and all linear maps to homomorphisms. Or for topological spaces, by changing vector spaces to topological spaces and linear maps to continuous maps.

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