## ABSTRACT

The Johnson scheme, the Hamming scheme and the unit sphere are particularly interesting and significant polynomial spaces. One important property that they have in common is the existence of what we will call an “addition rule”. In this chapter we explain what this is, and why it is useful. We characterise the polynomial spaces for which the partition of Ω × Ω determined by ρ is an association scheme, and show that every Q-polynomial association scheme determines a finite polynomial space in which our addition rule holds. It follows that the theory of Q-polynomial spaces can be viewed as an extension of the theory of Q-polynomial association schemes.