= Solution
The <tautological bundle> $\mathcal O(-1)$ over <Complex projective space> has fibre over $[\ell]$ equal to the line $\ell\subset\mathbb C^{r+1}$. Define $\mathcal O(1)=\mathcal O(-1)^*$ and the <twisting sheaf on projective space>, viewed analytically as a <holomorphic line bundle>, by
$$
\mathcal O(d)=
\begin{cases}\mathcal O(1)^{\otimes d},&d>0,\\
\mathcal O,&d=0,\\
\mathcal O(-1)^{\otimes(-d)},&d<0.
\end{cases}
$$
For $\mathbb{CP}^1$, use coordinates $z=Z_1/Z_0$ and $w=Z_0/Z_1=1/z$. The tautological frames $(1,z)$ and $(w,1)$ differ by $z^{-1}$, so frames $e_0,e_1$ for $\mathcal O(d)$ satisfy $e_1=z^d e_0$. A global <holomorphic section> is therefore described by entire functions $s_0(z),s_1(w)$ with
$$
s_1(w)=w^d s_0(1/w).
$$
Write $s_0(z)=\sum_{k\geq0}a_kz^k$. Holomorphicity of $s_1$ at zero permits only the powers $w^{d-k}$ with $d-k\geq0$. For $d\geq0$, this says $s_0$ is a <polynomial> of degree at most $d$; for $d<0$, every coefficient must vanish. Equivalently, when $d\geq0$ the sections are <homogeneous polynomials> of degree $d$ in $Z_0,Z_1$. Thus
$$
\boxed{\dim_{\mathbb C}H^0(\mathbb{CP}^1,\mathcal O(d))=
\begin{cases}d+1,&d\geq0,\\0,&d<0.\end{cases}}
$$
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