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Caching and evaluation doesn’t make any kind of sense for mathematical functions. They’re just mappings.

Yeah, but it’s true in a sense because an element of a finite-dimensional vector space can be thought of as a function from a finite set into a field.

Only if the space has a canonical basis.

Indeed, many linear algebra textbooks define a tuple of real numbers as a function f: {1,...,n} -> R.

yep, because of this one can do O(1) sigmoidals in float16 nnets with a 64K-word table.