The conformal bootstrap aims to systematically constrain CFTs based on crossing symmetry and
unitarity. In this talk I will introduce a new approach to extract information from the crossing symmetry
sum rules, based on the construction of linear functionals with certain positivity properties. I show
these functionals allow us to derive a class of optimal bounds on CFT data, and also act as an ideal basis
for obtaining other bounds numerically. Furthemore I will argue that special extremal solutions to
crossing form a basis for the crossing equation, with the functionals living in the dual space, providing a
rigorous derivation of the Polyakov (Mellin) bootstrap in 1D.