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This paper is the first in a series devoted to the development of a rigorousrenormalisation group method for lattice field theories involving boson fields,fermion fields, or both. Our immediate motivation is a specific model,involving both boson and fermion fields, which arises as a representation ofthe continuous-time weakly self-avoiding walk. In this paper, we define normedalgebras suitable for a renormalisation group analysis, and develop methods forperforming analysis on these algebras. We also develop the theory of Gaussianintegration on these normed algebras, and prove estimates for Gaussianintegrals. The concepts and results developed here provide a foundation for thecontinuation of the method presented in subsequent papers in the series.

A renormalisation group method. I. Gaussian integration and normed algebras