TITLE: Intuition behind tangent vectors as maps from $C^\infty$ functions to $\mathbb{R}$
QUESTION [2 upvotes]: In Wald's General Relativity, he writes:

Thus on a manifold $M$ let $\mathscr{F}$ denote the collection of $C^\infty$ functions from $M$ into $\mathbb{R}$. We define a tangent vector $v$ at point $p \in M$ to be a map $v : \mathscr{F} \to \mathbb{R}$ which (1) is linear and (2) obeys the Leibnitz rule.

Unfortunately I have very minimal background in the language of differential geometry and don't find this very intuitive; a clearer explanation would be really helpful.

REPLY [2 votes]: Have a look at the Wikipedia article Tangent space. The maps which define your tangent vectors are called derivations. So look in the cited article for the definition of tangent space in terms of derivations. But I suggest you to read the whole article. If you want to get deeper insights, also compare Bröcker, Jänich "Introduction to differential topology", where 3 different definitions of the tangent space are mentioned and proved to be equivalent. Have fun!
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