From linear to metric functional analysis

Significance

Metric geometry is a considerable extension of Riemannian geometry that, in recent decades, has proven very useful. A newer direction described in this article can moreover be viewed as an extension of functional analysis. I try to illustrate why these more recent concepts and results will be of importance, potentially generating new ideas in applied mathematics, for example, deep learning or evolution dynamics. An important objective of this note is therefore also to try to attract the attention of scientists in other disciplines. An extension of the contraction mapping principle and a recent fundamental theorem on the behavior of the composition of randomly selected transformation have vast potential for further applications.

Abstract

This article presents the beginning of a metric functional analysis. A major notion is metric functionals which extends that of horofunctions in metric geometry. Applications of the main tools are found in a wide variety of subjects such as random walks on groups, complex dynamics, surface topology, deep learning, evolution equations, and game theory, thus branching well outside of pure mathematics. In several cases, linear notions fail to describe linear phenomena that are naturally captured by metric concepts. An extension of the mean ergodic theorem testifies to this. A general metric fixed-point theorem is also proved.

Linearity is a fundamental notion in science, with concepts like derivatives and linear regression. It is also the main property in the foundational subject of functional analysis, which started developing with a shift in viewpoint from differential and integral equations, and their solutions, to linear operators and vector spaces of functions. The theory of Banach spaces is a further abstraction where the elements are thought of less as functions but rather just as points in a linear space. In this article, I would like to argue for a further step of generalization: forgetting the linearity of the space and instead focusing on merely the metric structure (coming from the norm in the Banach space case). This philosophy has been featured prominently in the Ribe program, initiated by J. Bourgain, J. Lindenstrauss, and others, with important applications (1, 2). This program, in particular, translates subtle geometric properties of the Banach spaces to metric spaces.

What I describe here is, while philosophically related, quite different; it is more basic and involves the operators too. A significant list of metric analogs for linear notions is recorded below. While this, in itself, is somewhat striking, what is more promising is that there are general tools that are remarkably powerful. In particular, I point out several phenomena within the linear theory that the metric notions describe better than what the linear notions can do.

More important still is the application to many nonlinear problems. Mathematics and its applications surprisingly abound with transformations preserving metrics. One instance that will be mentioned is found in deep learning (3), where maps are not linear, and this nonlinearity, imitating the functioning of a brain, is of decisive importance. Ball wrote already in ref. 4 that metric geometry has become a staple of mathematical computer science and the theory of algorithms. A survey of all of the uses of metric geometry is impossible, so the focus in this article is necessarily relatively narrow. But, already, what is discussed here, I think, will have an increasing impact also in applied mathematics and other sciences.

1. Tools in Metric Functional Analysis

Dieudonné wrote in ref. 5, p. 4 that, if one were to single out two crucial concepts in the development of functional analysis, they should be duality and spectral theory. Geodesics are the clear metric analog of lines; below, I will explain metric notions similar to the following ones from linear functional analysis related to duality:

• $•$continuous linear functionals,
• $•$the Hahn–Banach theorem,
• $•$weak topology, and
• $•$the Banach–Alaoglu theorem (weak compactness).

When one considers nonexpansive maps instead of operators, one moreover has analogs of the following spectral theoretic notions:

• $•$operator norm,
• $•$spectral radius, and
• $•$spectral theorem.

Not to forget is the most basic tool, the important contraction mapping principle, for example, used to establish the inverse mapping theorem and the existence of solutions to classes of equations. Both the spectral theorem and a fixed-point theorem established below serve as the replacement when the assumption in the contraction mapping principle does not hold.

I begin by explaining the first part of the analogy, which comes from the fundamental notion of a metric functional (which extends Busemann functions and horofunctions). This notion should be thought of as an analog of continuous linear functionals which, to some extent, is well recognized since Busemann, or, for example, refs. 6 and 7. I consider functions $X\to \mathbb{R}$ which vanish at an origin $x_0$ and are metric, in the sense that distances are not expanded. There are many such, but I take the most natural ones, those associated to points $x\in X$,

$$x\mapsto h_x\left(\cdot \right)≔d\left(\cdot ,x\right)-d\left(x_0,x\right).$$

Compare this with the Riesz representation theorem for continuous linear functionals in Hilbert spaces; they all correspond to vectors via the scalar product. One thus, in a sense, has a dual space of $X$, but one also has an injection of $X$ into this space. More formally, let $\left(X,d\right)$ be a metric space and fix a base point $x_0\in X$. Define

$$\mathrm{\Phi }:X\to {\mathbb{R}}^{\mathrm{X}}$$

via

$$x\mapsto h_x\left(\cdot \right)≔d\left(\cdot ,x\right)\hspace{0.17em}-\hspace{0.17em}d\left(x_0,x\right).$$

As the notation indicates, the topology I take here in the target space is pointwise convergence (in contrast to ref. 6). The map is continuous and injective. The following is a well-known consequence of the Tychonoff theorem and has been observed by many people; see, for example, refs. 8 and 9.

Proposition 1.1 (Metric Banach–Alaoglu). The space $\overline{\mathrm{\Phi }\left(X\right)}$ is a compact Hausdorff space.

The elements of $\overline{\mathrm{\Phi }\left(X\right)}$ are called metric functionals. In this way one has a weak topology with compactness properties for any metric space. Then there is the following:

Proposition 1.2 (Metric Hahn-Banach (10)). Let $\left(X,d\right)$ be a metric space with base point $x_0$ and $Y$ be a subset containing $x_0$. Then, for every $h\in \overline{Y}$, there is a metric functional $H\in \overline{X}$ which extends $h$ in the sense that $H{|}_Y=h$.

It is impossible to here enter any kind of survey on the use of metric functionals and horofunctions; a few references can be found in ref. 9, and here is one newer example: In ref. 11, the authors employ a certain metric and use that the horofunctions provide global viscosity solutions to the Hamilton–Jacobi equations.

I now turn to the analogs of operator theory. Let $\left(X,d\right)$ be a metric space and $f:X\to X$ be a nonexpansive map (i.e., a 1-Lipschitz map); that is, for all points $x,y\in X$, it holds that

$$d\left(f\left(x\right),f\left(y\right)\right)\hspace{0.17em}\le \hspace{0.17em}d\left(x,y\right).$$

Isometries constitute important examples. One defines the minimal displacement

$$d\left(f\right)\hspace{0.17em}=\hspace{0.17em}\underset{x}{inf}d\left(x,f\left(x\right)\right).$$

This serves as the analog to the operator norm, and the following is analogous to the spectral radius, called the translation number,

$$\tau \left(f\right)\hspace{0.17em}=\hspace{0.17em}\underset{n\to \infty }{lim}\frac{1}{n}d\left(x,f^n\left(x\right)\right).$$

Notice that $\tau \left(f\right)$ is independent of $x$ because, by nonexpansiveness, any two orbits stay at bounded distance from each other. This number exists by the Fekete lemma in view of the subadditivity coming from the triangle inequality and the 1-Lipschitz property. It has the tracial property $\tau \left(fg\right)=\tau \left(gf\right)$; see ref. 10. It is also easy to see the inequality $\tau \left(f\right)\le d\left(f\right)$.

The basic statement for such maps that complements the contraction mapping principle is:

Theorem 1.1. (Metric Spectral Theorem (8, 12)). Given a nonexpansive map $f:\left(X,d\right)\to \left(X,d\right)$, there exists $h\in \overline{X}$ such that

$$h\left(f^k\left(x_0\right)\right)\hspace{0.17em}\le \hspace{0.17em}-\tau \left(f\right)k$$

for all $k>0$, and, for any $x\in X$,

$$\underset{n\to \infty }{lim}-\frac{1}{n}h\left(f^n\left(x\right)\right)\hspace{0.17em}=\hspace{0.17em}\tau \left(f\right).$$

In case $X$ is a Banach space (or has a weak version of nonpositive curvature), $\tau \left(f\right)=d\left(f\right)$, and, moreover, $h\left(f\left(x\right)\right)\le h\left(x\right)-\tau \left(f\right)$ holds for every $x\in X$.

Metrics spaces are found all over mathematics, and their applications are both vast and rich. Traditionally, in differential geometry or geometric group theory, one deals with spaces satisfying some curvature condition (negative, nonpositive, nonnegative, positive, etc.), and, most often, locally compact spaces. Here I use no such condition for the fundamental facts. This includes many examples which do not have such properties in general (Hilbert metrics, Kobayashi metrics, Thurston’s Teichmüller distance, Hofer’s metric, etc.).

Gouëzel and I proved in ref. 13 a significant extension of this metric spectral statement, that generalizes Oseledets’ multiplicative ergodic theorem, as well as my earlier work with Karlsson and Margulis (14) and Karlsson and Ledrappier (15). Furstenberg asked in a seminal paper from 1963 (16) whether there is an extension of the law of large numbers to an arbitrary noncommutative group. Theorem 1.2 below provides an affirmative answer to this, since groups admit invariant metrics (whenever point-set topology allows it). I now explain the details.

Let $\left(\mathrm{\Omega },\mu \right)$ be a measure space with $\mu \left(\mathrm{\Omega }\right)=1$, and let $T:\mathrm{\Omega }\to \mathrm{\Omega }$ be a measure-preserving map that one assumes is ergodic. Let $g:\mathrm{\Omega }\to \mathrm{\Omega }$ be a measurable map into a semigroup G of nonexpansive maps of a metric space $X$. (For details about measurability, see ref. 13.) The following composition is called an ergodic cocycle:

$$u\left(n,\omega \right)\hspace{0.17em}=\hspace{0.17em}g\left(\omega \right)g\left(T\omega \right)\dots g\left(T^{n-1}\omega \right).$$

In addition, one needs to assume that the cocycle is integrable, which means that the integral of $d\left(g\left(\omega \right)x,x\right)$ over $\mathrm{\Omega }$ is finite, a condition which is independent of $x$.

Theorem 1.2 (Ergodic Theorem for Noncommuting Random Products (13)). Given an integrable ergodic cocycle $u\left(n,\omega \right)$ of nonexpansive maps of a metric space $\left(X,d\right)$, there exists a.s. a metric functional $h$ of $X$ such that

$$\underset{n\to \infty }{lim}-\frac{1}{n}h\left(u\left(n,\omega \right)x\right)\hspace{0.17em}=\hspace{0.17em}\underset{n\to \infty }{lim}\frac{1}{n}d\left(x,u\left(n,\omega \right)x\right).$$

An ergodic cocycle is intuitively a product of randomly selected maps, for example, selected independently and identically distributed (i.i.d.). Theorem 1.1 is the case without randomness: $g\left(\omega \right)=f$ and $u\left(n,\omega \right)=f^n$. That the latter limit in the theorem exists a.s. is a well-known direct consequence of Kingman’s subadditive ergodic theorem; the new part is the directional part as captured by a metric functional (for isometries, this statement was proved in ref. 15). Despite the question in ref. 16, I think it was unexpected that such a general statement existed. As Ledrappier and I showed in ref. 17, this can be applied to random walks on infinite groups even without knowing anything about the metric functionals of the given Cayley graph, for example, to establish that random walks on finitely generated groups of subexponential growth without homomorphism onto $\mathbb{Z}$ must have sublinear drift in word length.

2. When Linear Notions Fail

The Mean Ergodic Theorem.

The mean ergodic theorem of Carleman–von Neumann–Riesz from the 1930s (18) asserts that, for any Hilbert space operator $U$ of norm at most one, it holds that

$$\frac{1}{n}\sum _{k=0}^{n-1}{U}^{k}v\to w,$$ [1]

where $w$ is the projection of $v$ onto the subspace of $U$-invariant vectors. The metric spectral theorem above applied to the map $f\left(x\right)≔Ux+v$ gives more information than the above statement, namely, that there is a metric functional $h$ such that

$$h\left(Ux+v\right)\le h\left(x\right)-\tau$$

for all $x$ in the Hilbert space. This holds equally well in any Banach space, and, when iterated applied to $x=0$, it gives the following:

Theorem 2.1. (Mean Ergodic Theorem in Banach Spaces). For any operator $U$ of norm at most one of a Banach space and vector $v$, there is a metric functional $h$ such that

$$h\left(\sum _{k=0}^{n-1}{U}^{k}v\right)\hspace{0.17em}\le \hspace{0.17em}-n\tau ,$$

for all $n\ge 1$. Here $\tau =\underset{x}{inf}\Vert Ux+v-x\Vert$.

This implies the mean ergodic theorem recalled above for Hilbert spaces in view of the fact that metric functionals of this type are linear; see ref. 9 for details. It is well known that the usual mean ergodic theorem [1] does not hold in general Banach spaces, for example, with $U$ as the right shift on either ${\ell }^1$ or ${\ell }^{\infty }$. In contrast, Theorem 2.1 always holds. In the special case of $X=l^1$, no metric functional is identically zero (19), so this theorem gives nontrivial information for any operator $U$ with norm at most one and vector $v$, even when $\tau =0$.

A Metric Fixed-Point Theorem.

Schauder’s fixed-point theorem has found important applications in analysis. Compactness is a crucial ingredient for it to hold. For nonexpansive maps, one could hope for more, but Alspach (20) found an isometry of a weakly compact convex subset of $L^1\left[\mathrm{0,1}\right]$ without fixed point. The problem of whether such a map existed had remained open since it was raised by Kirk (21) and Browder (22). Gutiérrez noticed, in ref. 23, that the orbit of Alspach’s example, in fact, converges to a metric functional. More generally, this phenomenon can be explained by the following statement:

Theorem 2.2 (Metric Fixed-Point Theorem). Let $C$ be a convex set of a Banach space. Let $T:C\to C$ be an affine isometry that is surjective. Assume that $\tau \left(T\right)=0$. Then $T$ fixes a point in the metric compactification $\overline{C}$.

Proof: Let

$$y_n=\frac{1}{n}\left(x+Tx+\dots +T^{n-1}x\right),$$

which belongs to $C$ if $x\in C$. Consider a weak metric limit point $h$ of $h_{y_n}$ (with some $x_0\in C$ as chosen base point). Since $T$ is an isometry of $C$ onto itself, the action of $T$ is given by

$$\left(Th\right)\left(z\right)\hspace{0.17em}=\hspace{0.17em}h\left(T^{-1}z\right)-h\left(T^{-1}{x}_0\right).$$

Now, since

$$\begin{array}{l}\hfill h_{y_n}\left(T^{-1}z\right)-h_{y_n}\left(T^{-1}{x}_0\right)\hspace{0.17em}=\\ \hfill ‖\frac{1}{n}\left(x+Tx+\dots +T^{n-1}x\right)-T^{-1}z‖\\ \hfill -‖\frac{1}{n}\left(x+Tx+\dots +T^{n-1}x\right)-T^{-1}{x}_0‖\\ \hfill =‖\frac{1}{n}\left(Tx+T^{2}x+\dots +T^{n}x\right)-z‖\\ \hfill -‖\frac{1}{n}\left(Tx+T^{2}x+\dots +T^{n}x\right)-x_0‖,\end{array}$$

and $\tau \left(T\right)=\underset{n\to \infty }{lim}‖T^{n}x‖/n=0$, the difference of this expression above and $h_{y_n}\left(z\right)-h_{y_n}\left(x_0\right)$ disappears in the limit in view of the triangle inequality. Therefore $Th=h$. $\square$

In the special case that $C$ is bounded, then, obviously, $\tau \left(T\right)=0$, and the statement applies to Alspach’s isometry, thereby providing the missing fixed point as it were. The theorem is a good complement to Theorem 1.1, since the latter is especially informative in the case $\tau >0$. It can also be compared with ref. 24 that deals with $L^1$ in a more classical way.

Local Spectral Theory.

Beurling (25) and Gelfand (26) observed around 1940 that the spectral radius can be calculated by

$$\rho \left(A\right)\hspace{0.17em}=\hspace{0.17em}\underset{n\to \infty }{lim}{‖A^n‖}^{1/n},$$

which is the analog of the translation length $\tau$. One has the simple inequality $\rho \left(A\right)\le ‖A‖$, to compare with $\tau \left(f\right)\le d\left(f\right)$ in the metric setting. For a given vector $v$, one may ask about

$$\underset{n\to \infty }{lim}{‖A^{n}v‖}^{1/n}.$$

This limit may, on the other hand, not exist: An example can be given in ${\ell }^2$, and $A$ is a combination of a shift and a diagonal operator, having two exponents each alternating in longer and longer stretches, making the behavior different for various periods of $n$. See, for example, ref. 27 for details. This limits what one can hope for in terms of multiplicative ergodic theorems, discussed below, and, once more, is in contrast to Theorems 1.1 and 1.2 that hold for any metric space and any nonexpansive map.

Asymptotics of Nonexpansive Maps.

An example of Mertens in ref. 28 (example 3.2) shows that the metric functional from Theorem 1.1 describes the orbit much better than any linear functional. The map is the plane with the ${\ell }^1$ metric, where the points flow with constant speed to the vertical axis, and points on the vertical axis flow upward with the same speed. The map can be taken to be the time 1 map of this flow.

Another illustration is the following taken from ref. 29, and similar to Beals’ counterexample cited in ref. 22. Consider the nonexpansive map $T:{\ell }^1\left(\mathbb{N}\right)\to {\ell }^1\left(\mathbb{N}\right)$ defined by $T\left(x_1,x_2,\dots \right)=\left(1,x_1,x_2,\dots \right)$. So the orbit of zero is $T^{n}0=\left(\mathrm{1,1,1},\dots ,\mathrm{1,0,0},\dots \right)$. The map clearly has no fixed points in ${\ell }^1\left(\mathbb{N}\right)$, nor does

$$\frac{T^{n}x}{‖T^{n}x‖}$$

have any weak limit point. On the other hand, $h_{T^{n}0}$ converges to the metric functional

$$h\left(x\right)\hspace{0.17em}=\hspace{0.17em}\sum _{i=1}^{\infty }\left|x_i-1\right|-1$$

as $n\to \infty$. Moreover, $h\left(T^{n}0\right)\to -\infty$. This describes well the asymptotic behavior of the iterates of $T$.

The Invariant Subspace Problem.

One of the most notoriously hard problems in operator theory is to determine whether there is always a nontrivial closed invariant subspace. It is still open for separable complex Hilbert spaces of infinite dimension. For Banach spaces, there are counterexamples, first found by Enflo (30), and later Read (31) even found such cases for ${\ell }^1$. As is remarked in ref. 29, if there is a nontrivial continuous linear functional $f$ such that

$$f\left(Ux+v\right)\hspace{0.17em}\le \hspace{0.17em}f\left(x\right)$$

for some $v$ and every $x$, then $\ker f$ is a nontrivial closed linear subspace for $U$. Theorem 1.1 shows, in the case of ${\ell }^1$ and $‖U‖\le 1$ (obviously not a restriction in the search for invariant subspaces), that there is, for any $v\in {\ell }^1$, a nontrivial metric functional $h$ (in view of ref. 19) such that

$$h\left(Ux+v\right)\le h\left(x\right)$$

for every $x$. Read’s example thus shows that the same statement cannot hold with a continuous linear functional.

3. Nonlinear Applications

Here follows a small selection of applications and potential applications of Theorems 1.1 and 1.2. Note that, in each context, there is no metric present at the outset.

Wolff–Denjoy Theorems.

Historically, the first case of Theorem 1.1 is the Wolff–Denjoy theorem obtained in a series of research notes published by the French Academy of Sciences in 1926. This classic result asserts that any holomorphic self-map $f$ of the unit disk $D$ either fixes a point, or its iterates $f^k\left(z\right)$ converge to a boundary point, and the horodisks centered at that point are $f$-invariant sets. Indeed, in view of the Schwarz–Pick lemma asserting that $f$ is nonexpansive with respect to the negatively curved Poincaré metric, one has, from Theorem 1.1, that there is a metric functional (horofunction) $h$ such that $h\left(f^k\left(z\right)\right)\le h\left(z\right)-kd\left(f\right)$ for every $k>0$ and $z\in D$. This is a slight improvement on the classical theorem because of the appearance of $d\left(f\right)$. The literature on extensions of the Wolff–Denjoy theorem is vast; see the recent contribution (32) containing many further references. Note that Kobayashi’s pseudometric makes every holomorphic map in one or several dimensions nonexpansive. Therefore, Theorems 1.1 and 1.2 apply. Examples of random products of holomorphic maps are found in the theory of continued fraction expansion; see also refs. 13, 33, and 34.

Thurston’s Spectral Theorem.

Let $M$ be an oriented closed surface of genus $g\ge 2$. Let $\mathcal{S}$ denote the isotopy classes of simple closed curves on $M$ not isotopically trivial. For a Riemannian metric $\rho$ on $M$, let $l_{\rho }\left(\beta \right)$ be the infimum of the length of curves isotopic to $\beta$. In a seminal preprint from 1976 (35), Thurston announced the existence of eigenvalues or Lyapunov exponents for surface homeomorphisms. Using my approach in ref. 36, employing Thurston’s Lipschitz metric on Teichmüller space in combination with ref. 37, the isometry case of Theorem 1.2, Horbez (38) could show the full random extension of Thurston’s spectral theorem:

Theorem 3.1 (38). For any i.i.d. composition with finite first moment of homeomorphisms $f_n=g_n{g}_{n-1}\dots g_1$, there is a finite set $1\le {\lambda }_1<{\lambda }_2<\dots <{\lambda }_K$ such that, for any $\alpha \in \mathcal{S}$, there is a ${\lambda }_i$ such that, for any Riemannian metric $\rho$,

$$\underset{n\to \infty }{lim}{l}_{\rho }{\left(f_n\alpha \right)}^{1/n}={\lambda }_i.$$

Each exponent ${\lambda }_k$ has an associated subsurface, and the ${\lambda }_i$ in the conclusion is the maximal one among the subsurfaces that $\alpha$ intersects.

The analogy with matrices is clear. Since the metric methods are rather soft, one could maybe find extensions to diffeomorphisms of compact manifolds in higher dimensions. One possibility could be symplectomorphisms using Hofer’s metric.

Deep Neural Networks.

The theoretical understanding of deep learning is very limited, in particular the role of the number of layers, called depth, and why the fits tend to be better with increased depth and not to overfit as in traditional statistics. In a paper with Avelin (3), we associate metric spaces on which the layer maps in deep neural networks act as nonexpansive maps, for several of the most common choices in practice. The application of this that we develop is to study the so-called deep limits, that is, when the number of layers tend to infinity. After a determination of the metric functionals, one can use Theorem 1.2. This shows certain convergent behaviors, sometimes thought of as undesirable, for example, in the deep Gaussian processes which is the case studied previously (39–41) from this point of view. Or perhaps it is an indication of a regularity that prevents overfitting.

Nonlinear Perron–Frobenius Theory.

Applied mathematics is rich with nonexpansive maps, particularly on cones. The most standard cone is the one with all vectors with positive entries. An excellent reference on this topic is the book by Lemmens and Nussbaum (42). A notable example is the operator introduced in the celebrated paper (43), now called the Shapley operator, concerning zero-sum stochastic games. It is a nonexpansive map in the supremum norm and without fixed points. Thus the metric spectral theorem is applicable; see refs. 8 and 44. In ref. 8, Gaubert and Vigeral proved, using this approach, the existence of an initial state that is the best for the maximizing player in the long run. It remains to be seen whether a random version instead using Theorem 1.2 is of interest from a game theoretic perspective.

Evolution Partial Differential Equations.

The need for multiplicative ergodic theorems for operators has been expressed in influential articles (45–47). For example, in one approach to the Navier–Stokes equation, related evolution equations for climate modeling and stochastic partial differential equations, the dynamics takes place in infinite dimensional Hilbert spaces. In recent years, there has been progress on such operator multiplicative ergodic theorems; see, for example, refs. 14, 27, 48–50 and references therein. Let me remark that the metric approach is taken in refs. 13, 14, and 50, and, in ref. 14, for Hilbert–Schmidt operators, the result is a stronger, more uniform version than what Oseledets theorem would suggest or ref. 45 provides for compact operators. The theorems in ref. 13 may ultimately lead to the most general results (compare with the remark on local spectral theory above). The metric approach uses the isometric action of invertible operators on associated symmetric spaces of positive operators. The first case of this is very classical: $2\times 2$ real matrices of determinant one acting by Möbius transformations on the upper half-plane leaving the hyperbolic distance invariant, which, for many questions about this group, is easier to study than the linear action on ${\mathbb{R}}^2$.

Data Availability

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