  In this article we will give a affirmative answer to Gilbert's conjecture on
Hardy spaces of Clifford analytic functions in upper half-space of
$\mathbb{R}^8$. It depends on a explicit construction of Spinor space
$\mathcal{R}_8$ and Clifford algebra $Cl_8$ by octonion algbra. What's more ,
it gives us an associative way to octonionic analytic function theory. And the
similar question has been discussed in Octonionic Hardy space in upper-half
space, some classical results about octonionic analytic functions have been
reformulated, too.
