Summary: Square-integrable octonion-valued function spaces on \(\mathbb{R}^3\) and \(\mathbb{R}^7\) are decomposed into the direct sum of octonion Hardy and conjugate Hardy spaces, and square-integrable octonion function spaces on the upper half spaces \(\mathbb{R}^4_+\) and \(\mathbb{R}^8_+\) are decomposed into infinity direct sum of subspaces in which the first components are just the octonion Bergman spaces. The octonion Hardy spaces on \(\mathbb{R}^3\) and \(\mathbb{R}^7\) are characterized.