Definition of sub-bundle

Question

In these notes (Example 2.19) it is said that a sub-bundle $E'$ of $E$ should be a bundle $E' \subset E$ such that $E/E'$ is also a vector bundle. I feel a bit confused as I can't imagine bundles $E' \subset E$ with $E/E'$ is not a vector bundle.

Answer 1

A vector bundle can be seen as a locally free sheaf. But there exists injective morphisms $\mathcal{F}\rightarrow\mathcal{G}$ between locally free sheaves such that the quotient is not locally free, hence not a vector bundle. For this reason, $\mathcal{F}$ is not a sub-bundle.

An example is the following : let $X=\mathbb{P}^1$ and $P$ be any point. Then we have a short exact sequence $$ 0\longrightarrow\mathcal{O}_{\mathbb{P}^1}(-1)\longrightarrow\mathcal{O}_{\mathbb{P}^1}\longrightarrow\mathcal{O}_P\longrightarrow 0$$ The locally free sheaf $\mathcal{O}_{\mathbb{P}^1}(-1)$ is indeed a subsheaf of the locally free sheaf $\mathcal{O}_{\mathbb{P}^1}$ but not a sub-bundle since the quotient is not locally free.