Characterization of linear normality

Question

A projective variety $X\subseteq\mathbb{P}^{n}$ is said to be linearly normal if the restriction map $$ H^{0}(\mathbb{P}^{n}, \mathcal{O}_{\mathbb{P}^{n}}(1))\rightarrow H^{0}(X,\mathcal{O}_{X}(1)) $$ is surjective. I have read that $X\subseteq\mathbb{P}^{n}$ fails to be linearly normal if and only if it is the linear projection of a nondegenerate embedding $X\subseteq \mathbb{P}^{n+1}$. Nevertheless, I don't get to understand this equivalence. I would appreciate if someone could explain it.

Answer 1

Why is this confusing, except that yo have written $n$ instead of $n+1$ above. If $X\subset\mathbb{P}^{n+1}$ is non-degenerate, then $\dim H^0(X,\mathcal{O}_X(1))\geq n+2$. If it can be projected to $\mathbb{P}^n$, the map $H^0(\mathcal{O}_{\mathbb{P}^n}(1))\to H^0(\mathcal{O}_X(1))$ can not be onto, since the former has dimension $n+1$ and the latter has dimension at least $n+2$.