For a ring and a set, the set of functions (to the underlying set of ) is itself naturally an associative algebra over , where addition and multiplication is given pointwise in by addition and multiplication in : for their sum is the function
their product is the function
and the ring inclusion is given by sending to the constant function with value .
More generally, in the context of (∞,1)-topos theory and higher algebra, there is a notion of function algebras on ∞-stacks.
If is a finite set or else if one restricts to functions that are non-vanishing only for finitely many elements in , then the algebra of functions with values in also forms the free module over generated by .
If is a set and is a commutative ring, then the pointwise multiplication on the function algebra is the Hadamard product on the function algebra.
Sending spaces to their suitable algebras of functions constitutes a basic duality operation that relates geometry and algebra. For more on this see at Isbell duality.
Last revised on August 21, 2024 at 02:08:12. See the history of this page for a list of all contributions to it.