Consider the Segre embedding $$ \mathbb P^n\times \mathbb P^m \hookrightarrow \mathbb P^N. $$

It seems to me that from the definition it is clear that

$$ H^0(\mathbb P^N, \mathscr O_{\mathbb P^N}(1)) = H^0(\mathbb P^n, \mathscr O_{\mathbb P^n}(1))\otimes H^0(\mathbb P^m, \mathscr O_{\mathbb P^m}(1)), $$

but is it obvious that

$$ H^0(\mathbb P^n\times \mathbb P^m, {\mathscr O_{\mathbb P^{N}}(1)}|_{\mathbb P^n\times \mathbb P^m}) = H^0(\mathbb P^n, \mathscr O_{\mathbb P^n}(1))\otimes H^0(\mathbb P^m, \mathscr O_{\mathbb P^m}(1))? $$

More generally:

Question: Is the Segre embedding projectively normal?

**EDIT** Note that strictly speaking the above condition is only linearly normal, I did mean to ask *projective normality*, which in this case follows from linear normality. (See J.C. Ottem's answer).