I am reading K-theory by Michael Atiyah, and I am confused with this definition.
Let $E$ be a vector bundle over space $X$, and $\Gamma(E)$ be the vector space of all sections of $E$. We say a subspace $V\subset\Gamma(E)$ is ample if the map $\varphi:X\times V\to E$, defined by $\varphi(x,s)=s(x)$, is surjective.
I don't quite understand the definition, as my intuition says if the map $\varphi$ is surjective, $V$ has to be $\Gamma(E)$. Can anyone help me please?
Consider the example where $E$ is the trivial line bundle $E = V\times \mathbb{C}$. Then $\Gamma(E)$ can be identified with smooth functions $V \to \mathbb{C}$; given such a function $f$, the corresponding section is $s_f$ defined by $s_f(x) = (x, f(x))$. Let $V \subset \Gamma(E)$ be the subspace of sections corresponding to constant functions. For any $(x, \alpha) \in E$, there exists $s \in V$ such that $\varphi(x, s) = s(x) = (x, \alpha)$ - just take $s$ to be the section associated to the constant function $\alpha$. So $V$ is ample but is far from all of $\Gamma(E)$.