Definition:Vector Space of Germs of Smooth Functions

Definition

Let $M$ be a smooth manifold with or without boundary.

Let $p \in M$.

Let $\GG_p$ denote the set $\map {C_p^\infty} M$ of all germs of smooth functions at $p$.

Let $\sqbrk f_p \in \GG_p$ denote the germ of $f$ at $p$ where $f$ is a smooth function.

Then $\struct {\GG_p, +_p, \cdot_p}_\R$ is a vector space over $\R$, in which the operations:

$+_p: \GG_p \times \GG_p \to \GG_p$

$\cdot_p: \R \times \GG_p \to \GG_p$

are defined as:

			vector addition:	\(\ds \forall \sqbrk f_p, \forall \sqbrk g_p \in \GG_p:\)		\(\ds \sqbrk f_p +_p \sqbrk g_p \)	\(\ds := \)	\(\ds \sqbrk {f + g}_p \)
			scalar multiplication:	\(\ds \forall c \in \R: \forall \sqbrk f_p \in \GG_p:\)		\(\ds c \cdot_p \sqbrk f_p \)	\(\ds := \)	\(\ds \sqbrk {c f}_p \)

where:

$f + g$ denote the pointwise addition of functions;

$c f$ denote the pointwise scalar multiplication of functions.

Although this article appears correct, it's inelegant. There has to be a better way of doing it. In particular: We could probably do well to introduce a more streamlined notation to replace $\map {C_p^\infty} M$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Vector Space of Germs of Smooth Functions is Vector Space

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): Chapter $3$: Tangent Vectors: $\S$ Alternative Definitions of the Tangent Space