A (complex, semi-definite) inner product space is a complex vector space equipped with a sesquilinear form which is conjugate symmetric, in the sense that for all , and non-negative in the sense that for all . By inspecting the non-negativity of for complex numbers , one obtains the Cauchy-Schwarz inequality

if one then defines , one then quickly concludes the triangle inequality

which then soon implies that is a semi-norm on . If we make the additional assumption that the inner product is positive definite, i.e. that whenever is non-zero, then this semi-norm becomes a norm. If is complete with respect to the metric induced by this norm, then 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 is replaced with a -ary inner product 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 , the Gowers box norms , and the Gowers-Host-Kra seminorms ; a more elementary example are the family of Lebesgue spaces 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 on a measure space , where the inner product takes the form

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.

— 1. Definition of a higher order Hilbert space —

Let be complex vector spaces. Then one can form the (algebraic) tensor product , which can be defined as the vector space spanned by formal tensor products , subject to the constraint that the tensor product is bilinear (i.e. that , , and similarly with the roles of and reversed). More generally, one can define the tensor product of any finite family of complex vector spaces .

Given a complex vector space , one can define its complex conjugate to be the set of formal conjugates of vectors in , with the vector space operations given by

The map is then an antilinear isomorphism from to . We adopt the convention that , thus is also an antilinear isomorphism from to . (One can work with real higher order Hilbert spaces instead of complex ones, in which case the conjugation symbols can be completely ignored.)

For inductive reasons, it is convenient to use finite sets of labels, rather than natural numbers , to index the order of the systems we will be studying. In any case, the cardinality of the set of labels will be the most important feature of this set.

Given a complex vector space and a finite set of labels, we form the tensor cube to be

where is the conjugation map , and when ; thus for instance , is spanned by tensor products with , is spanned by tensor products with , and so forth. (It would be better to order the four factors in a square pattern, rather than linearly as is done here, but we have used the inferior linear ordering here for typographical reasons.)

Given any finite set of labels and any , one can form an identification

by identifying a tensor product in with

where, for and , denotes the element of that agrees with on and equals on . We refer to this identification as , thus

is an isomorphism, and one can define the tensor product of two elements . Thus for instance, if and are elements of , then

using the linear ordering conventions used earlier. If we instead view as elements of rather than , then

A (semi-)definite inner product on a complex vector space can be viewed as a linear functional on obeying a conjugation symmetry and positive (semi-)definiteness property, defined on tensor products as . With this notation, the conjugation symmetry axiom becomes

and the positive semi-definiteness property becomes

with equality iff in the definite case.

Now we can define a higher order inner product space.

Definition 1 Let be a finite set of labels. A (semi-definite) inner product space of order is a complex vector space , together with a linear functional that obeys the following axiom: (Splitting axiom) For every , is a semi-definite classical inner product on , which we identify with using as mentioned above. We say that the inner product space is positive definite if one has whenever is non-zero. (Note from the splitting axiom that one already has the non-strict inequality. But the positive definiteness property is weaker than the assertion that each of the classical inner products)

For instance, if is the empty set, then an inner product space of order is just a complex vector space equipped with a linear functional from to (which one could interpret as an expectation or a trace, if one wished). If is a singleton set, then an inner product space of order is the same thing as a classical inner product space.

If , then an inner product space of order is a complex vector space equipped with a linear functional , which in particular gives rise to a quartisesquilinear (!) form

which is a classical inner product in two different ways, thus for instance we have

for and some classical inner product on , and similarly

for some classical inner product on .

— 2. Examples —

Let us now give the three major (and inter-related) examples of inner product spaces of higher order: the Gowers uniformity spaces, that arise in additive combinatorics; the Gowers box spaces, which arise in hypergraph regularity theory, and the Gowers-Host-Kra spaces, which arise in ergodic theory. We also remark on the much simpler example of the Lebesgue spaces of dyadic exponent.

The first example is the family of Gowers uniformity spaces , which we will define for simplicity on a finite additive group (one can also define this norm more generally on finite subsets of abelian groups, and probably also nilpotent groups, but we will not do so here). Here is a finite set of labels; in applications one usually sets , in which case one abbreviates as . The space is the space of all functions , and so can be canonically identified with the space of functions . To make into an inner product space of order , we define

where is the subgroup of consisting of the parallelopipeds

This is clearly a linear functional. To verify the splitting axiom, one observes the identity

for any and . The right-hand side is then a semi-definite classical inner product on ; the semi-definiteness becomes more apparent if one makes the substitution .

Specialising to tensor products, we obtain the Gowers inner product

Thus, for instance, when ,

The second example is the family of the (incomplete) Gowers box spaces , defined on a Cartesian product of a family of measure spaces indexed by a finite set . To avoid some minor technicalities regarding absolute integrability, we assume that all the measure spaces have finite measure (the theory also works in the -finite case, but we will not discuss this here). This space is the space of all bounded measurable functions (here, for technical reasons, it is best not to quotient out by almost everywhere equivalence until later in the theory). The tensor power can thus be identified with a subspace of (roughly speaking, this is the subspace of “elementary functions”). We can then define an inner product of order by the formula

for all , where and are integrated using product measure .

The verification of the splitting property is analogous to that for the Gowers uniformity spaces. Indeed, there is the identity

for all and , where , , and for . From this formula one can verify the inner product property without much trouble (the main difficulty here is simply in unpacking all the notation).

The third example is that of the (incomplete) Gowers-Host-Kra spaces . Here, is a probability space with an invertible measure-preserving shift , which of course induces a measure-preserving action of the integers on . (One can replace the integers in the discussion that follows by more general nilpotent amenable groups, but we will stick to integer actions for simplicity.) It is often convenient to also assume that the measure is ergodic, though this is not strictly required to define the semi-norms. The space here is ; the power is then a subspace of . One can define the Host-Kra measure on for any finite by the following recursive procedure. Firstly, when is empty, then is just . If instead is non-empty, then pick an element and view as the Cartesian product of with itself. The shift acts on , and thus acts diagonally on by acting on each component separately. It is not hard to show inductively from the construction that we are about to give that is invariant with respect to this diagonal shift, which we will call . The product -algebra has an invariant factor with respect to this shift. We then define to be the relative product of with itself relative to this invariant factor. One can show that this definition is independent of the choice of , and that the form

is an inner product of order ; see the paper of Host and Kra for details.

A final (and significantly simpler) example of a inner product space of order is the Lebesgue space on some measure space , with inner product

where is the diagonal embedding from to . For tensor products, this inner product takes the form

thus for instance when ,

We leave it as an exercise to the reader to show is indeed an inner product space of order . This example is (the completion of) the Gowers-Host-Kra space in the case when the shift is trivial.

We also remark that given an inner product space of some order , given some subset of , and given a fixed vector in , one can define a weighted inner product space of order by the formula

for all , where is embedded in by extension by zero and the tensor product on the right-hand side is defined in the obvious manner. One can check that this is indeed a weighted inner product space. This is a generalisation of the classical fact that every vector in an inner product space naturally defines a linear functional on . In the case of the Gowers uniformity spaces with , this construction takes to ; similarly for the Gowers box spaces.

— 3. Basic theory —

Let be an inner product space of order for some finite non-empty . The splitting axiom tells us that

for all , , and some inner product on . In particular one has

for all , as well as the classical Cauchy-Schwarz inequality

If we specialise this inequality to the tensor products

for various , one concludes that

where we write for some and . If we iterate this inequality once for each , we obtain the Cauchy-Schwarz-Gowers inequality

where

The quantity is clearly non-negative and homogeneous. We also have the Gowers triangle inequality

which makes a semi-norm (and in fact a norm, if the inner product space was positive definite). To see this inequality, we first raise both sides to the power :

The left-hand side can be expanded as

which after expanding out using linearity and the triangle inequality, can be bounded by

which by the Cauchy-Schwarz-Gowers inequality can be bounded in turn by

which can then be factored into as required.

Note that when is a singleton set, the above argument collapses to the usual derivation of the triangle inequality from the classical Cauchy-Schwarz inequality. It is also instructive to see how this collapses to one of the standard proofs of the triangle inequality for using a large number of applications of the Cauchy-Schwarz inequality.

In analogy with classical Hilbert spaces, one can define a Hilbert space of order to be an inner product space of order which is both positive definite and complete, so that the norm gives the structure of a Banach space. A typical example is for a finite abelian , which is the space of all functions with the norm

where is the Pontraygin dual of (i.e. the space of homomorphisms from to ) and is the Fourier transform. Thus we see that is a Hilbert space of order . More generally, for any measure space and any can be viewed as a Hilbert space of order .

The Gowers norms and Gowers-Host-Kra norms coincide in the model case when is a cyclic group with uniform measure and the standard shift . Also, the Gowers norms can be viewed as a special case of the box norms via the identity

where is the summation operation .

Just as classical inner product spaces can be made positive definite by quotienting out the norm zero elements, and then made into a classical Hilbert space by metric completion, inner product spaces of any order can also be made positive definite and completed. One can apply this procedure for instance to obtain the completed Gowers box spaces and the completed Gowers-Host-Kra spaces (which become when the shift is trivial). These spaces are related, but not equal, to their Lebesgue counterparts ; for instance for the Gowers-Host-Kra spaces in the ergodic setting, a repeated application of Young’s inequality reveals the inequalities

and so contains a (quotient) of .

The null space of the Gowers-Host-Kra norm in in the ergodic case is quite interesting; it turns out to be the space of bounded measurable functions whose conditional expectation on the characteristic factor of order of vanishes; in particular, becomes a dense subspace of , embedded injectively. It is a highly non-trivial and useful result, first obtained by Host and Kra), that is the inverse limit of all nilsystem factors of step at most ; this is the ergodic counterpart of the inverse conjecture for the Gowers norms.

— 4. The category of higher order inner product spaces —

The higher order Hilbert spaces are related to each other via Hölder’s inequality; the pointwise product of two functions is in , the product of two functions is in , and so forth. Furthermore, the inner products on all of these spaces are can be connected to each other via the pointwise product.

We can generalise this concept, giving the class of inner product spaces (of arbitrary orders) the structure of a category.

Definition 2 Let be finite sets, and let , be inner product spaces of order respectively. An isometry from to is a linear map which preserves the inner product in the sense that where is the obvious concatenation map from to . Given an isometry from to , and an isometry from to for some , one can form the composition by the formula and extending by linearity; one can verify that this continues to be an isometry, and that the class of inner product spaces of arbitrary order together with isomorphisms form a category.

When is a singleton set, the above concept collapses to the classical notion of an isometry for inner product spaces. Of course, one could specialise to the subcategory of higher order Hilbert spaces if desired. The inner product on a higher order inner product space can now be interpreted as an isometry from that space to the space (viewed as an inner product space of order ), and is the unique such isometry; in the language of category theory, this space becomes the terminal object of the category.

A model example of an isometry is the sesquilinear product map , which is an isometry from to for any . For the Gowers-Host-Kra norms, the map is an isometry from to for any and .

To see analogous isometries for the Gowers uniformity norms, one has to generalise these norms to the “non-ergodic” setting when one does not average the shift parameter over the entire group , but on a subgroup . Specifically, for finite additive groups and functions with , define the local Gowers inner product

By foliating into cosets of , one can express this local Gowers inner product as an amalgam of the ordinary Gowers inner product and a Lebesgue inner product. For instance, one has the identity

We define the inner product space to be the space of functions from to with the above inner product. Given any , we can then create an isometry from to by defining

(This isometry does not ostensibly depend on , except through the labels of the inner product of the target space of the isometry.)

One can obtain analogous isometries for the Gowers box norms after similarly generalising to “non-ergodic” settings; we leave this as an exercise to the interested reader.

Actually, the “derivative maps” from inner product spaces of order to those of order can be constructed abstractly. Indeed, one can view as an inner product space of order with the inner product defined on tensor products by

and then the map is an isometry. One can iterate this construction and obtain a cubic complex of inner product spaces

of order for each , together with a commuting system of derivative isometries from to for each .

Conversely, one can use cubic complexes to build higher order inner product spaces:

Proposition 3 Let be a finite set. For each , suppose that we have a vector space equipped with a -sesquilinear form and suppose that for each one has a sesquilinear product obeying the compatibility conditions whenever for all . Suppose also that the form is a classical inner product on for every . Then for each , is an inner product space of order , and the maps become isometries.

This proposition is established by an easy induction on the cardinality of . Note that we do not require the derivative maps to commute with each other, although this is almost always the case in applications.

[Update, May 20: added section on cubic complexes.]