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Pointwise Operation

A pointwise operation on the function space Y^X is induced by an n-ary operation omega:Y^n->Y through

omega(f_1,...,f_n)(x)=omega[f_1(x),...,f_n(x)].

Thus, for example, pointwise addition and multiplication of real-valued functions are defined by (f+g)(x)=f(x)+g(x) and (fg)(x)=f(x)g(x). Algebraic identities satisfied by omega are inherited by its pointwise extension, so function spaces naturally acquire structures such as groups, rings, and vector spaces.

See also

Function Space, Operation, Pointwise Convergence

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References

Lang, S. Algebra, 3rd ed. New York: Springer-Verlag, 2002.

Cite this as:

Weisstein, Eric W. "Pointwise Operation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PointwiseOperation.html

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