The T _z-Seminorm

We introduce the T_z-seminorm, which is used in subsequent chapters to measure the size of the nonperturbative coordinate of the renormalisation group map. We define the seminorm, prove its important product property, show how it can be used to obtain bounds on derivatives, and explain in which sense the seminorm of a Gaussian expectation is bounded by the expectation of the seminorm. Good properties of the seminorm with respect to exponentiation and Taylor expansion are developed; the latter is an essential ingredient in our proof of the crucial contraction property of the renormalisation group map. We conclude with some estimates on polynomials for later use.