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For a positive integer $k$ and a function $f:G\to \mathbb{C}$ on a finite abelian group, with uniform expectations, definewhere $\mathcal{C}$ is complex conjugation. The alternating conjugations measure agreement around additive cubes. The expression is nonnegative; for $k\ge 2$ its $2^k$th root is a norm. At $k=1$ it is only the seminorm $|\mathbb{E}f|$.

With the multiplicative derivative ${\partial }_{h}f(x)=f(x+h)\overline{f(x)}$,Repeated multiplicative derivatives remove low-degree polynomial phases: a unit-modulus phase of degree less than $k$ has $U^k$ norm one. The Gowers U2 norm is the fourth moment of Fourier coefficients on a finite abelian group and detects linear correlations; the Gowers U3 norm controls four-term arithmetic progression counts through the generalized von Neumann inequality for four-term progressions.^[ref]

Let $X$ be a nonempty finite set, and let $\mathcal{F}$ be a nonempty, uniformly bounded family of real-valued functions on $X$. With the normalized inner product, defineUniform boundedness makes the supremum finite. Absolute homogeneity and the triangle inequality follow from the same properties of each linear functional. This is a norm exactly when the linear span of $\mathcal{F}$ is all of ${\mathbb{R}}^X$, so no nonzero $g$ is orthogonal to every test.^[ref]

Separating the underlying points of $X$ is insufficient. On two points, the single test $f=(1,2)$ takes distinct values, yet the nonzero vector $g=(2,-1)$ has zero test seminorm. The tests must detect differences between whole functions.