We develop a pathwise integration theory based on Faber-Schauder energy spaces. The approach replaces the classical H\"older-Young and finite-variation Young conditions by dyadic summability conditions expressed in terms of Faber--Schauder coefficients. On the normalized interval $[0,1]$, these conditions define Banach spaces $E^p$, which we call Faber-Schauder energy spaces. For $p,q>1$ satisfying $1/p+1/q\ge1$, we prove that every pair $f \in E^p$ and $g \in E^q$ admits a continuous pathwise integral $I_{f,g}$, constructed from dyadic left Riemann sums. We call $I_{f,g}$ the Faber-Schauder integral, and show that it depends boundedly and bilinearly on $(f,g)$ in the corresponding energy norms. The integral satisfies additivity, integration by parts, and a dyadic Young-Lo\`eve estimate. It is also the uniform limit of classical Riemann-Stieltjes integrals of finite Faber-Schauder approximations. The Faber-Schauder integral agrees with the classical Young integral whenever the latter is available, but also applies to deterministic and Gaussian examples for which neither the H\"older-Young condition nor the finite-variation Young condition can be verified. In this sense, it provides a Faber-Schauder coefficient-based extension of Young's framework.