Problem 2.3 (Vector space structure on derivations at a point)

Question

Vector space structure on derivations at a point
Let $D$ and $D^{\prime}$ be derivations at $p$ in $\mathbb{R}^n$, and $c \in \mathbb{R}$. Prove that
\begin{enumerate}[label=(\alph*)]
\item the sum $D+D^{\prime}$ is a derivation at $p$.
\item the scalar multiple $c D$ is a derivation at $p$.
\end{enumerate}

Toğrul Topal · Accepted Answer

Let $D_1$ and $D_2$ be two point-derivations at $p \in \mathbb{R}^n$. Our goal is to show that $D_1 + D_2$ and $cD_1$ are also point-derivations at $p$, i.e., both are linear and satisfy Leibniz rule.
ewline

Linearity is obvious for both scalar multiple and the sum.
ewline

We check the Leibniz rule for the sum:
$$
\begin{aligned}
\forall f,g \in C^\infty_p: \quad \left(D_1+D_2ight)(f g) & =D_1(f g)+D_2(f g) \
& =D_1(f) g(p)+D_1(g) f(p)+D_2(f) g(p)+D_2(g) f(p) \
& =\left(D_1(f)+D_2(f)ight) g(p)+\left(D_1(g)+D_2(g)ight) f(p) \
& =\left(D_1+D_2ight)(f) g(p)+\left(D_1+D_2ight)(g) f(p)
\end{aligned}
$$

Now we check the Leibniz rule for the scalar multiple of $D$.

$$
c D(f g)=c(D(f g))=c(D(f) g(p)+D(g) f(p))=c D(f) g(p)+c D(g) f(p),
$$

Problem 2.3 (Vector space structure on derivations at a point)

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Problem 2.3 (Vector space structure on derivations at a point)

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