# Short exact sequence of vector bundles vs locally free sheaves

If I have a short exact sequence of vector bundles $$0\rightarrow E' \rightarrow E \rightarrow E'' \rightarrow 0$$ then it splits, which mean that $E = E'\oplus E''$? So if I have a short exact sequence of locally free sheaves $$0\rightarrow \mathcal{F}' \rightarrow \mathcal{F}\rightarrow \mathcal{F}'' \rightarrow 0$$ then is it also true that the sequence splits and I have $\mathcal{F} = \mathcal{F}'\oplus\mathcal{F}''$?

In general, the answer is no. For example, the Euler sequence on $$\mathbb{CP}^n$$, $$0 \to \mathcal O_{\mathbb P^n} \to \mathcal O_{\mathbb P^n}(1)^{\oplus (n + 1)} \to \mathcal T_{\mathbb P^n} \to 0,$$ splits when viewed as a SES of smooth vector bundles, but it is not possible to find a splitting that respect the holomorphic structure. (See here - I particularly like Ben's answer.)