  We introduce a new approach to the study of the crossing equation for CFTs in
the presence of a boundary. We argue that there is a basis for this equation
related to the generalized free field solution. The dual basis is a set of
linear functionals which act on the crossing equation to give a set of sum
rules on the boundary CFT data: the functional bootstrap equations. We show
these equations are essentially equivalent to a Polyakov-type approach to the
bootstrap of BCFTs, and show how to fix the so-called contact term ambiguity in
that context. Finally, the functional bootstrap equations diagonalize
perturbation theory around generalized free fields, which we use to recover the
Wilson-Fisher BCFT data in the $\epsilon$-expansion to order $\epsilon^2$.
