Formal calculus MOC

Formal sums over a vector space

Let be a vector space over . Then the space of formal sums in indeterminate with coëfficients in and exponents in are denoted , #m/def/fcalc with elements of the form . is itself a vector space with addition and scaling defined pointwise.

If , we have the following useful subspaces¹

• has exponents in only, and is called Taylor series over ;
• has exponents in only, and is called Laurent series over ;
• has exponents in only and finitely many terms, and is called polynomials over ; and
• has exponents in only and finitely many terms, and is called Laurent polynomials over

Given we define , and for evaluation may be defined similarly. We have two well-defined bilinear multiplication maps

See also

• Formal sums over endomorphisms
• Formal sums over a Lie algebra
• Degree operator on formal sums over a vector space

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