Retrieving "Real Vector Space" from the archives

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1. Smooth Manifold

   Linked via "real vector space"

   For any point $p \in M$, the tangent space $T_pM$ is defined as the set of all derivations on the algebra of smooth real-valued functions $C^\infty(U)$ defined in a neighborhood $U$ of $p$. A derivation $X$ at $p$ is a linear map satisfying the Leibniz rule: $X(fg) = fX(g) + gX(f)$ for smooth functions $f, g$.
   If $M$ is $n$-dimensional, $TpM$ is an $n$-dimensional real vector space. The collection of all tangent spaces $\{TpM\}_{p \in M}$ defines the tangent bundle $TM$, which is a $2n$-dimensional [fib…