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Let V be the set of all functions

f: $\mathbb{R}$ $\to$ $\mathbb{R}$

such that the n-th derivative "$f^n$" of $f$ exists for all integers $n$.

We define a non standard addition ⊕ on V by setting $f$ ⊕ $g$ = $f'$ + $g'$ where $f'$ and $g'$ are the derivatives of $f$ and $g$ respectively.

V together with non-standard additon, scalar multiplication and $0$ is not a vector space.

Which vector space axioms does V not satisfy?

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  • $\begingroup$ Uhm... I wonder which ones it does satisfy. $\endgroup$
    – user239203
    Feb 2, 2021 at 9:08
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    $\begingroup$ You could talk about derivative of $f:\mathbb R^2 \to \mathbb R^2$, but then $f' = (\partial f_i / \partial x_j)$ which is a $2\times 2$ matrix not a map $\mathbb R^2\to \mathbb R^2$. I'm confused :/ $\endgroup$
    – AlvinL
    Feb 2, 2021 at 9:09
  • $\begingroup$ Probably the derivatives should be replaced with the differentials, which are linear maps: $f\oplus g=\mathrm df+\mathrm dg$. $\endgroup$
    – Bernard
    Feb 2, 2021 at 9:36
  • $\begingroup$ @AlvinLepik Oh sorry I meant f: $\mathbb{R}$ $\to$ $\mathbb{R}$. $\endgroup$
    – user878169
    Feb 2, 2021 at 9:53

1 Answer 1

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Assuming the multiplication with scalar is defined in the obvious way. We would need the identity $(\lambda+\mu)f = \lambda f \oplus\mu f$, but in general $$\lambda f \oplus \mu f = \lambda f' + \mu f' \neq \lambda f + \mu f. $$

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