In proving $\mathfrak X(U)$ is a $C^{\infty}U$-module, for an open subset $U$ of $\mathbb R^n$ my book defines scalar multiplication of smooth vector fields $U$ by smooth functions on $U$ as

$$[fX]_p := f(p)X_p$$

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I could not find a definition for addition. Is it $[X+Y]_p := X_p + Y_p$ ? Otherwise, with what definition do we prove the "Abelian group" part of showing $\mathfrak X(U)$ is a $C^{\infty}U$-module?

My book is An Introduction to Manifolds by Loring W. Tu.

  • 6
    $\begingroup$ It is pointwise addition, just as you say. $\endgroup$ Jan 26, 2019 at 9:46
  • 1
    $\begingroup$ @CharlieFrohman Thanks! $\endgroup$
    – user636532
    Jan 26, 2019 at 9:46

1 Answer 1


As Charlie pointed in the comments, it is just your regular pointwise addition: $$(X+Y)_p=X_p+Y_p$$


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