  We study a renormalization group (RG) map for tensor networks that include
two-dimensional lattice spin systems such as the Ising model. Numerical studies
of such RG maps have been quite successful at reproducing the known critical
behavior. In those numerical studies the RG map must be truncated to keep the
dimension of the legs of the tensors bounded. Our tensors act on an
infinite-dimensional Hilbert space, and our RG map does not involve any
truncations. Our RG map has a trivial fixed point which represents the
high-temperature fixed point. We prove that if we start with a tensor that is
close to this fixed point tensor, then the iterates of the RG map converge in
the Hilbert-Schmidt norm to the fixed point tensor. It is important to
emphasize that this statement is not true for the simplest tensor network RG
map in which one simply contracts four copies of the tensor to define the
renormalized tensor. The linearization of this simple RG map about the fixed
point is not a contraction due to the presence of so-called CDL tensors. Our
work provides a first step towards the important problem of the rigorous study
of RG maps for tensor networks in a neighborhood of the critical point.
