Problem 2.2 (Algebra structure on $C^{\infty}_p$)

Question

Algebra structure on $C_p^{\infty}$ Define carefully addition, multiplication, and scalar multiplication in $C_p^{\infty}$. Prove that addition in $C_p^{\infty}$ is commutative.

Toğrul Topal · Accepted Answer

Recall that $C^{\infty}_p$ is the set of equivalence classes, called germs, $[(U,f)]$ where $U$ is an open neighbourhood of the point $p$ and $f:U 	o \mathbb{R}$ is a smooth function. We say that $(U,f)$ is equivalent to $(V,g)$ iff there is an open set $W \subseteq U \cap V$ also containing $p$ such that $festriction_W = gestriction_W$.
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Let $(U,f)$ and $(V,g)$ be twe representatives of germs $[(U,f)], [(V,g)] \in C^{\infty}_p$. We define addition via
\begin{equation*}
    [(U,f)] + [(V,g)] := [(U \cap V, (f+g)estriction_W],
\end{equation*}
multiplication via
\begin{equation*}
    [(U,f)] \cdot [(V,g)] := [(U \cap V, (f \cdot g)estriction_W],
\end{equation*}
and scalar multiplication via
\begin{equation*}
    c \cdot [(U,f)] := [(U, c \cdot f],
\end{equation*}
It is easy to verify that these operations satisfy algebra axioms. Commutativity follows by commutativity of set intersection ($U \cap V = V \cap U$) and commutativity of addition on the codomains ($(f + g)estriction_W = (g + f)estriction_W$). Furthermore, both operations are well-defined in that they do not depend on the choice of representatives $(U,f)$ and $(V,g)$.

Problem 2.2 (Algebra structure on $C^{\infty}_p$)

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Problem 2.2 (Algebra structure on $C^{\infty}_p$)

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