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A (complex, semi-definite) inner product space is a complex vector space ${V}$ equipped with a sesquilinear form ${\langle, \rangle: V \times V \rightarrow {\bf C}}$ which is conjugate symmetric, in the sense that ${\langle w, v \rangle = \overline{\langle v, w \rangle}}$ for all ${v,w \in V}$, and non-negative in the sense that ${\langle v, v \rangle \geq 0}$ for all ${v \in V}$. By inspecting the non-negativity of ${\langle v+\lambda w, v+\lambda w\rangle}$ for complex numbers ${\lambda \in {\bf C}}$, one obtains the Cauchy-Schwarz inequality

$\displaystyle |\langle v, w \rangle| \leq |\langle v, v \rangle|^{1/2} |\langle w, w \rangle|^{1/2};$

if one then defines ${\|v\| := |\langle v, v \rangle|^{1/2}}$, one then quickly concludes the triangle inequality

$\displaystyle \|v + w \| \leq \|v\| + \|w\|$

which then soon implies that ${\| \|}$ is a semi-norm on ${V}$. If we make the additional assumption that the inner product ${\langle,\rangle}$ is positive definite, i.e. that ${\langle v, v \rangle > 0}$ whenever ${v}$ is non-zero, then this semi-norm becomes a norm. If ${V}$ is complete with respect to the metric ${d(v,w) := \|v-w\|}$ induced by this norm, then ${V}$ is called a Hilbert space.

The above material is extremely standard, and can be found in any graduate real analysis course; I myself covered it here. But what is perhaps less well known (except inside the fields of additive combinatorics and ergodic theory) is that the above theory of classical Hilbert spaces is just the first case of a hierarchy of higher order Hilbert spaces, in which the binary inner product ${f, g \mapsto \langle f, g \rangle}$ is replaced with a ${2^d}$-ary inner product ${(f_\omega)_{\omega \in \{0,1\}^d} \mapsto \langle (f_\omega)_{\omega \in \{0,1\}^d}}$ that obeys an appropriate generalisation of the conjugate symmetry, sesquilinearity, and positive semi-definiteness axioms. Such inner products then obey a higher order Cauchy-Schwarz inequality, known as the Cauchy-Schwarz-Gowers inequality, and then also obey a triangle inequality and become semi-norms (or norms, if the inner product was non-degenerate). Examples of such norms and spaces include the Gowers uniformity norms ${\| \|_{U^d(G)}}$, the Gowers box norms ${\| \|_{\Box^d(X_1 \times \ldots \times X_d)}}$, and the Gowers-Host-Kra seminorms ${\| \|_{U^d(X)}}$; a more elementary example are the family of Lebesgue spaces ${L^{2^d}(X)}$ when the exponent is a power of two. They play a central role in modern additive combinatorics and to certain aspects of ergodic theory, particularly those relating to Szemerédi’s theorem (or its ergodic counterpart, the Furstenberg multiple recurrence theorem); they also arise in the regularity theory of hypergraphs (which is not unrelated to the other two topics).

A simple example to keep in mind here is the order two Hilbert space ${L^4(X)}$ on a measure space ${X = (X,{\mathcal B},\mu)}$, where the inner product takes the form

$\displaystyle \langle f_{00}, f_{01}, f_{10}, f_{11} \rangle_{L^4(X)} := \int_X f_{00}(x) \overline{f_{01}(x)} \overline{f_{10}(x)} f_{11}(x)\ d\mu(x).$

In this brief note I would like to set out the abstract theory of such higher order Hilbert spaces. This is not new material, being already implicit in the breakthrough papers of Gowers and Host-Kra, but I just wanted to emphasise the fact that the material is abstract, and is not particularly tied to any explicit choice of norm so long as a certain axiom are satisfied. (Also, I wanted to write things down so that I would not have to reconstruct this formalism again in the future.) Unfortunately, the notation is quite heavy and the abstract axiom is a little strange; it may be that there is a better way to formulate things. In this particular case it does seem that a concrete approach is significantly clearer, but abstraction is at least possible.

Note: the discussion below is likely to be comprehensible only to readers who already have some exposure to the Gowers norms.