Examples and definition of multilinear mappings on infinite products

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Can anybody give me examples of multilinear mappings (trivial or non trivial) on an infinite product space $∏_{i\in I} X_i$, where $I$ is an infinite set and $X_i$'s are objects, say vector spaces.

I know the multilinear mappings are defined on finite products ${\times}{^n}_{i=1} X_i$ in the literature, but I'm interested to know how people extended them for infinite products. So any reference for the definition of multilinear mappings on infinite products?