Elements of Statistical Learning - question on p. 12
You are correct, at p+1 it is returning to talk about the K=1 case again. p inputs produce 1 output. The linear function $\beta$ you fit maps p dimensional X to a p dimensional subspace of the p+1 dimensional space that includes the 'output' and it's specifically just a hyperplane. Try plotting $z = 3x + 2y$ in a graphing calculator for a quick intuition of what single output from p=2 inputs feels like.