Formal sums over a vector space

Let $𝑉$ be a vector space over $𝕂$ . Then the space of formal sums in indeterminate $𝑧$ with coëfficients in $𝑉$ and exponents in $𝕂$ are denoted $𝑉 {𝑧}$ , fcalc with elements of the form $\sum 𝑛 \in 𝕂 𝑣 𝑛 𝑧 𝑛$ . $𝑉 {𝑧}$ is itself a vector space with addition and scaling defined pointwise.

\sum 𝑛 \in 𝕂 𝑣 𝑛 𝑧 𝑛 + \sum 𝑛 \in 𝕂 𝑤 𝑛 𝑧 𝑛 = \sum 𝑛 \in 𝕂 (𝑣 𝑛 + 𝑤 𝑛) 𝑧 𝑛 𝛼 \sum 𝑛 \in 𝕂 𝑣 𝑛 𝑧 𝑛 = \sum 𝑛 \in 𝕂 𝛼 𝑣 𝑛 𝑧 𝑛

If $c h a r 𝕂 = 0$ , we have the following useful subspaces¹

$𝑉 [[𝑧]]$ has exponents in $ℕ 0$ only, and is called Taylor series over $𝑉$ ;
$𝑉 [[𝑧, 𝑧 - 1]]$ has exponents in $ℤ$ only, and is called Laurent series over $𝑉$ ;
$𝑉 [𝑧] = 𝑉 \otimes 𝕂 [𝑧]$ has exponents in $ℕ 0$ only and finitely many terms, and is called polynomials over $𝑉$ ; and
$𝑉 [𝑧, 𝑧 - 1] = 𝑉 \otimes 𝕂 [𝑧, 𝑧 - 1]$ has exponents in $ℤ$ only and finitely many terms, and is called Laurent polynomials over $𝑉$

Given $𝑣 (𝑧) = \sum 𝑛 \in ℤ 𝑣 𝑛 𝑧 𝑛 \in 𝑉 [[𝑧, 𝑧 - 1]]$ we define $𝑣 (𝛼 𝑧) = \sum 𝑛 \in ℤ 𝛼 𝑛 𝑣 𝑛 𝑧 𝑛$ , and for $𝑣 (𝑧) \in 𝑉 [𝑧, 𝑧 - 1]$ evaluation may be defined similarly. We have two well-defined bilinear multiplication maps

𝑉 [𝑧, 𝑧 - 1] \times 𝕂 {𝑧} \to 𝑉 {𝑧} 𝑉 [𝑧, 𝑧 - 1] \times 𝕂 [[𝑧, 𝑧 - 1]] \to 𝑉 [[𝑧, 𝑧 - 1]]

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Formal sums over a vector space

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Table of Contents

Formal sums over a vector space

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