# Vector field Axioms

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?

• Uhm... I wonder which ones it does satisfy.
– user239203
Commented Feb 2, 2021 at 9:08
• 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 :/ Commented Feb 2, 2021 at 9:09
• Probably the derivatives should be replaced with the differentials, which are linear maps: $f\oplus g=\mathrm df+\mathrm dg$. Commented Feb 2, 2021 at 9:36
• @AlvinLepik Oh sorry I meant f: $\mathbb{R}$ $\to$ $\mathbb{R}$.
– user878169
Commented Feb 2, 2021 at 9:53

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 $$\lambda f \oplus \mu f = \lambda f' + \mu f' \neq \lambda f + \mu f.$$