  We describe the volume dependence of matrix elements of local fields to all
orders in inverse powers of the volume (i.e. only neglecting contributions that
decay exponentially with volume). Using the scaling Lee-Yang model and the
Ising model in a magnetic field as testing ground, we compare them to matrix
elements extracted in finite volume using truncated conformal space approach to
exact form factors obtained using the bootstrap method. We obtain solid
confirmation for the form factor bootstrap, which is different from all
previously available tests in that it is a non-perturbative and direct
comparison of exact form factors to multi-particle matrix elements of local
operators, computed from the Hamiltonian formulation of the quantum field
theory. We also demonstrate that combining form factor bootstrap and truncated
conformal space is an effective method for evaluating finite volume form
factors in integrable field theories over the whole range in volume.
