An infinite dimensional version of the intermediate value theorem

Let f=I-k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {f}= I-k$$\end{document} be a compact vector field of class C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} on a real Hilbert space H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}$$\end{document}. In the spirit of Bolzano’s Theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}) and Kronecker (in Rk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^k$$\end{document}), we prove an existence result for the zeros of f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {f}$$\end{document} in the open unit ball B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}$$\end{document} of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}$$\end{document}. Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction f|S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {f}|_\mathbb {S}$$\end{document} of f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {f}$$\end{document} to the boundary S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}$$\end{document} of B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}$$\end{document}. As an extension of this, but not as a consequence, we obtain as well an Intermediate Value Theorem whose statement needs the topological degree. Such a result implies the following easily comprehensible, nontrivial, generalization of the classical Intermediate Value Theorem: If a half-line with extremeq∉f(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \notin \mathfrak {f}(\mathbb {S})$$\end{document}intersects transversally the functionf|S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {f}|_\mathbb {S}$$\end{document}for only one point of S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}$$\end{document}, then any value of the connected component ofH\f(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}{\setminus }\mathfrak {f}(\mathbb {S})$$\end{document}containingqis assumed byf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {f}$$\end{document}inB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}$$\end{document}. In particular, such a component is bounded.


Introduction
Let H be a real Hilbert space. Denote by B the open unit ball of H and by B its closure, also called the unit disk of H. The boundary ∂B of B is the unit sphere of H, hereafter denoted by S.
Let f : H → H be a compact vector field; namely, a map of the type I − k, where I is the identity of H and k is a compact map, meaning that k is As we shall see, an easily understandable hypothesis implying the condition | deg bf ((f − q) ∂ )| = 0 is the following: There exists a half-line with extreme q whose intersection with f(S) is transverse and its preimage under f| S consists of an odd number of points.
A final example will show that the converse implication is not true.
It is worth pointing out that many authors addressed the problem of defining an integer-valued degree for Fredholm maps; see, e.g., [4,21,25] for a comprehensive discussion. Among them we cite Fitzpatrick, Pejsachowicz and Rabier, who defined in [13] a notion of degree for C 2 Fredholm maps between real Banach manifolds, based on a concept of orientation for this class of maps; such a degree has been extended to the C 1 case in [22]. This notion of orientation is different from the one that we follow here, introduced by the first and third author in [6]. We think that a result analogous to Theorem 6.5 holds true in this context; however we are not able to prove it because of some technical difficulties.

Preliminaries
Here we expose some notation and preliminaries that we will need later.
Recall that a continuous map between metrizable spaces is said to be proper if the pre-image of any compact set is a compact set. It is easy to check that proper maps are closed, in the sense that the image of any closed set is a closed set.
Let, hereafter, H and K be two real Hilbert spaces. By L(H, K) we will denote the Banach space of the bounded linear operators from H into K, endowed with the usual operator norm. For simplicity, the notation L(H) stands for L(H, H). By Iso(H, K) we shall mean the open subset of L(H, K) of the invertible operators, and we will write GL(H) instead of Iso(H, H).
We recall that an operator L ∈ L(H, K) is said to be Fredholm (see e.g. [24]) if both its kernel, Ker L, and its cokernel, coKer L = K/L(H), are finite dimensional. In this case its index is the integer ind L = dim(Ker L) − dim(coKer L).
Obviously, if L ∈ L(H, K) is invertible, then it is Fredholm of index 0. Moreover, any operator in L(R k , R s ) is Fredholm with index k − s.
By Φ(H, K) we denote the subset of L(H, K) of the Fredholm operators. Given n ∈ Z, Φ n (H, K) stands for {L ∈ Φ(H, K) : ind L = n}. In particular, Φ(H) and Φ n (H) designate Φ(H, H) and Φ n (H, H), respectively. Unless otherwise stated, with the symbol I we mean the identity operator acting on any vector space.
If L ∈ L(H) has the property that I −L ∈ F(H), we shall say that L is a det-admissible operator. The symbol A(H) will stand for the affine subspace Since the identity on H is Fredholm of index 0, from Property 5 of Proposition 2.1 one gets that A(H) is a subset of Φ 0 (H).
In [16], the determinant of an operator L ∈ A(H) is defined as det L := det L| X , where L| X is the restriction of L (as domain and as codomain) to any finite dimensional subspace X of H containing the image of I − L, with the understanding that det L| X = 1 if X = {0}.
Here are some fundamental properties of the determinant.
See, for example, [9] for a discussion about other properties. The easy proof of the following remark is left to the reader.
Suppose that, according to this splitting, L ∈ L(H) can be represented in a block matrix form as where I 22 is the identity operator on Y. Then L ∈ A(H) and det L = det L 11 .
Given L ∈ Φ 0 (H, K), in [6], an operator K ∈ F(H, K) was called a corrector of L if L + K ∈ Iso(H, K). Since the word "corrector" is misleading (an invertible operator need not to be corrected), we will use the more appropriate word companion.
Any L ∈ Iso(H, K) has a natural companion: the trivial element of the space L(H, K). This fact was of fundamental importance for two concepts of orientation introduced in [6] and, consequently, for the construction of the bf-degree.
Given any L ∈ Φ 0 (H, K), let C(L) denote the subset of the vector space F(H, K) of the companions of L. Whatever is L, invertible or non-invertible, this set is nonempty and, according to the following definition, it admits a partition in exactly two equivalence classes (see [6] for details).

Definition 3.3 (Equivalence relation of companions)
. Two companions K 1 and K 2 of an operator L ∈ Φ 0 (H, K) are equivalent (more precisely, Lequivalent) if the determinant of the det-admissible operator (L + K 2 ) −1 (L + K 1 ) is positive. This is an equivalence relation on C(L) with two equivalence classes.
The following concept, introduced in [6], is based on Definition 3.3. Some special algebraic orientations are in order. Definition 3.6 deals with the finite dimensional case. Definition 3.6 (Associated alg-orientation of a linear operator ). Let H and K have the same finite dimension. Assume that they are oriented up to an inversion of both the orientations (or, equivalently, assume that H×K is oriented). Then any L ∈ L(H, K) admits the alg-orientation which is associated to the orientations of H and K: the one given by considering as a positive companion of L any K ∈ C(L) such that L + K is orientation preserving.
Recall that, if dim H < ∞ and L ∈ L(H), by sign L one simply means the sign of det L. More generally, if H and K have the same finite dimension and are oriented, then the (classical) sign of L is defined as follows (see, for example, [19]): Going beyond the finite dimensional context, we introduce the following concept of sign of an alg-oriented operator, called here bf-sign in order to distinguish it from the above classical notion. Definition 3.7 (bf-sign of an alg-oriented operator ). Let L ∈ Φ 0 (H, K) be alg-oriented. Its bf-sign is the integer if L is invertible and naturally alg-oriented, −1 if L is invertible and not naturally alg-oriented.
The easy proof of the following remark is left to the reader.
Remark 3.8. Assume that H and K have the same finite dimension and are oriented, then, given L ∈ L(H, K), one has sign L = sign bf L, provided that L has the associated alg-orientation.

Topological orientation
We now extend the concept of orientation to nonlinear maps (see [6,7] for more details).
The basic fact is that the alg-orientation of an operatorĽ ∈ Φ 0 (H, K) induces an alg-orientation to the operators in a neighborhood ofĽ. In fact, since Iso(H, K) and Φ 0 (H, K) are open in L(H, K), any companion ofĽ remains a companion of all L sufficiently close toĽ. . Let X be a topological space and Γ : X → Φ 0 (H, K) a continuous map. A function ω that to any x ∈ X assigns an alg-orientation ω(x) of Γ(x) is called a topological orientation of Γ (top-orientation for short) provided it is locally constant in the following sense: ifx ∈ X and K ∈ ω(x), then K ∈ ω(x) for all x in a neighborhood ofx. The map Γ is called top-orientable if it admits a top-orientation, and top-oriented if a top-orientation has been chosen.
Definition 3.10 (Pull-back of an orientation in the flat case). Let X and Y be topological spaces. If σ : X → Y and Γ : Y → Φ 0 (H, K) are continuous maps, then any orientation ω of Γ induces an orientation ω * of the composite map Γ • σ by putting ω * (x) = ω(Γ(σ(x))) for all x ∈ X . We will say that the orientation ω * is the pull-back of ω or, informally, that the orientation ω * is induced on Γ • σ by ω.
From Definition 3.10 one gets that Φ 0 (H, K) is locally top-orientable and, if A ⊆ Φ 0 (H, K) is top-orientable, then so is any subset of A, as it is any continuous map Γ : X → A. In particular, any constant map from a topological space into Φ 0 (H, K) is top-orientable.
If H and K have the same finite dimension, then Φ 0 (H, K), which in this case coincides with L(H, K), is top-orientable. In fact, assuming that H and K are oriented, one can assign, according to Definition 3.6, the associated alg-orientation to any operator of L(H, K). One can check that this turns out to be a top-orientation.
However, Φ 0 (H, K) could be not top-orientable in the infinite dimensional context. In fact, a surprising result of N. Kuiper (see [10,18]) asserts that, if H is infinite dimensional and separable, then the linear group GL(H) is contractible. Actually, in the context of Banach spaces, in [7,Theorem 3.15] it is shown that, given a Banach space F, if GL(F) is connected, then the open subset Φ 0 (F) of L(F) is as well connected but not top-orientable.
Hereafter, byΦ 0 (H, K) we shall mean the set of the alg-oriented Φ 0operators acting from H to K. Namely, Remark 3.11. (A two-fold covering space in the flat case) Definition 3.9 implies that the setΦ 0 (H, K) can be endowed with the topology which makes the natural projection a 2-fold covering space (see [7] for details). Therefore, given a continuous map Γ : X → Φ 0 (H, K) defined on a topological space, the second component ω of a lift of Γ is a top-orientation of Γ, and the function may be regarded, abusing of the notation introduced in Definition 3.9, as a top-oriented map of Φ 0 -operators.
The covering space P :Φ 0 (H, K) → Φ 0 (H, K) is useful for dealing with the top-orientability, as well as with the top-orientation, of continuous maps into Φ 0 (H, K); this is the case, for example, of the differential df : Since, by definition, a simply connected topological space is assumed to be path connected, from Remark 3.11 and the theory of covering spaces, one gets that, if X is simply connected and locally path connected, then Γ : X → Φ 0 (H, K) admits exactly two top-orientations. Moreover, ifx ∈ X and α is any of the two alg-orientations of Γ(x), then there exists one and only one top-orientation ω of Γ such that ω(x) = α.
Particular attention should be paid to the special convex subset LS(H) of L(H) consisting of the compact linear perturbations of the identity, that we shall call Leray-Schauder subset of L(H). Notice that Property 5 of Proposition 2.1 implies that LS(H) is contained in Φ 0 (H). Therefore, since it is simply connected and locally path connected, the following definition makes sense.

Definition 3.12 (Standard top-orientation of the Leray-Schauder subset of L(H)). The unique top-orientation ω of LS(H) such that ω(I)
is the natural alg-orientation of the identity (see Definition 3.5) will be called standard.
We now recall the concept of top-orientation for maps between manifolds. To this end, we need an analogous 2-fold covering space for the more general case in which the Hilbert spaces H and K are replaced by two manifolds M and N . For this purpose we define two sets: the base space ; and the total spacê We need to define the topologies on these two sets in order to make the natural projection P + : (p, q, L, α) → (p, q, L) a covering space.
The topology on the base space Φ + 0 (M, N ) is defined as follows. Let ϕ : U → H and ψ : V → K be two charts of M and N , respectively. Then, there exists a bijective correspondence between the subset . In fact, one can check that the map is a bijection. Thus, in Φ + 0 (U, V ) we consider the topology which makes I a homeomorphism; obtaining, in this way, a neighborhood basis of any element (p,q,Ľ) ∈ Φ + 0 (U, V ). Since ϕ and ψ are arbitrary, we get a topology on Φ + 0 (M, N ). Similarly, we define the topology on the total spaceΦ + 0 (M, N ). Let ϕ : U → H and ψ : V → K be two charts as above, and consider the bijection K) is a topological space, the topology onΦ + 0 (M, N ) can be defined with exactly the same argument used for Φ + 0 (M, N ), and this implies the following result.  N ). The following orientability criterion can be deduced from the theory of covering spaces. Recall that a simply connected space is assumed to be path connected.  N ) be a continuous map defined on a topological space. If X is simply connected and locally path connected, then Γ + admits exactly two top-orientations. Moreover, if the restriction Γ + | E of Γ + to a path connected subset E of X has a top-orientationω, then there exists a unique top-orientationω of Γ + whose restriction to E coincides withω.
Let us now introduce the notion of orientation for nonlinear Fredholm maps between Hilbert spaces. The following important definition is propaedeutic to the more important concept of orientation for nonlinear Fredholm maps between manifolds, that we will introduce in Definition 3.18 below.  A special and important case of a Φ 0 -map defined on an open subset W of H is a C 1 compact vector field. Namely, a C 1 map f : W → H with the property that k := I − f is a compact map; that is, a map sending bounded sets into relatively compact sets (see, e.g., [11,Sect. 8]). It is known that, for any p ∈ W , the differential df p belongs to the convex subset LS(H) of Φ 0 (H) consisting of the compact linear perturbation of the identity operator I. Therefore, the following definition makes sense. An operator L ∈ Φ 0 (H, K) can also be regarded as a C 1 map from W = H into K. Therefore, for L we have two different notions of orientation: the alg-orientation (see Definition 3.4) and the top-orientation (see Definition 3.16). Since dL : H → K is the constant map dL p = L for all p ∈ H, we shall tacitly assume that the two possible orientations, if given, coincide. More precisely: if C + (L) is the class of positive companions for L, then it is as well for dL p for all p ∈ H. The next result is a direct consequence of Definition 3.18 and Proposition 3.15. Recall that a simply connected space is assumed to be path connected and observe that any manifold is locally path connected.

Proposition 3.19. (Orientability criterion for Φ 0 -maps between manifolds)
Let f : M → N be a Φ 0 -map between two manifolds. If M is simply connected, then f admits exactly two top-orientations.
A simple example of a not top-orientable Φ 0 -map is given by a constant map f from the 2-dimensional real projective space P 2 into R 2 (see [7]). This is due to the fact that even dimensional real projective spaces are nonorientable. Incidentally, we observe that, although f is constant, this is not the case for the continuous map (Natural) A special top-orientation of a Φ 0 -map between manifolds is the natural one, which makes sense whenever f : M → N is a diffeomorphism (or, more generally, a local diffeomorphism): given any p ∈ M, according to Definition 3.5, one assigns the natural alg-orientation to the differential df p .
Vol. 25 (2023) An infinite dimensional version Page 11 of 25 70 (Associated) Given a C 1 map f : M → N between two finite dimensional oriented manifolds of the same dimension, one can assign to f the top-orientation which is associated to the orientations of M and N : to any differential df p one assigns the alg-orientation which is associated to the orientations of T p M and T f (p) N , according to Definition 3.6.
(Canonical) If f is a self-map of a connected, orientable, finite dimensional manifold M, one can assign to f the canonical top-orientation: the one which is associated to any orientation of M, provided that it is the same as domain and as codomain of f . Now we define the concept of Φ 0 -homotopy between two manifolds, as well as the notion of its top-orientation. Denoting by ∂ 1 H : (p, t) → d(H t ) p the partial differential of H with respect to the first variable, a top-orientation of H is, according to Definition 3.14, a top-orientation of the map A Φ 0 -homotopy H is said to be top-orientable if it admits a top-orientation, and top-oriented if a top-orientation has been chosen.
Let ω be a top-orientation of a Φ 0 -homotopy H from M to N . Then, given any parameter t ∈ [0, 1], ω t := ω(·, t) is a top-orientation of the partial map H t , called the partial top-orientation of H at t. Conversely, one has the following consequence of the theory of covering spaces (see [6,7]).

Proposition 3.22. (Transport of the top-orientations for Φ 0 -maps) Assume that a partial map H t : M → N of a Φ 0 -homotopy H has a top-orientation α.
Then, there exists one and only one top-orientation ω of H whose partial toporientation ω t coincides with α. In particular, if two maps from M to N are Φ 0 -homotopic, then they are both top-orientable or both not top-orientable.
From Proposition 3.22 we deduce that any self-map of a manifold M which is Φ 0 -homotopic to the identity is top-orientable. Indeed, being a diffeomorphism, the identity admits the natural top-orientation (see Definition 3.20). This agrees with a well known fact: if a finite dimensional manifold is non-orientable, then it is not contractible. Proposition 3.22 could be extended to a wider class than the Φ 0 -homotopies: Φ 0 -families of maps between manifolds, just by replacing the parameter space [0, 1] with a simply connected and locally path connected topological space.

Topological degree
Here we summarize the main concepts related to the degree introduced in [6] for maps between real Banach manifolds, hereafter called bf-degree to distinguish it from other classical degrees, such as the Brouwer degree, Brdegree, and the Leray-Schauder degree, LS-degree (see [5,7,8] for additional details).
By an axiomatic approach, as in the work of Amann-Weiss [3] regarding the uniqueness of the Leray-Schauder degree, in [8] it is shown that the bf-degree is the only possible integer valued function satisfying three fundamental properties called Normalization, Additivity and Homotopy Invariance (see below for precise statements).
To be more specific, we need to define the domain of the bf-degree function. Given a top-oriented Φ 0 -map f : M → N between two manifolds, an open (possibly empty) subset U of M, and a target value q ∈ N , the triple (f, U, q) is said to be bf-admissible for the bf-degree if U ∩ f −1 (q) is compact. From the axiomatic point of view, the bf-degree is an integer valued function, deg bf , defined on the class of all the bf-admissible triples, that satisfies the following three fundamental properties: • Remark 4.1. Notice that the above Homotopy Invariance Property applies whenever H is a proper map.
Other useful properties of the bf-degree can be deduced from the three fundamental ones (see [8] for details). One of them is the Another one is the Observe that, if (f, U, q) satisfies the above two properties, then f −1 (q)∩ U is a compact set which does not intersect ∂U . Hereafter such a triple will be called strictly bf-admissible.
The following straightforward consequence of the Homotopy Invariance Property holds for the strictly bf-admissible triples.
• A simple example of a bf-admissible triple which is not strictly bfadmissible is given by (exp, R, q), where q ∈ R is arbitrary and the function exp is assumed to be canonically top-oriented (see Definition 3.20); in fact, the integer valued function deg bf (exp, R, ·) is discontinuous at 0 ∈ R.
Suppose that a strictly bf-admissible triple (f, M, q) satisfies the following additional property: • the codomain N of f is connected.
Then, the above Continuous Dependence Property implies that deg bf (f, M, q) does not depend on the target q ∈ N . Therefore, as for the Brouwer degree of a self-map of a compact, connected, orientable manifold, we will adopt a short notation. Notice that, in the above notation, the open set M in which the degree is considered is not mentioned. In fact, this is unnecessary since, in this case, M is the whole domain of the map f , and the domain of a function is included in its formal definition as a triple: domain, codomain and graph. As a simple example of degree for strictly bf-admissible triples consider a complex polynomial P of (algebraic) degree n > 0 and regard it as a selfmap of R 2 . Then P is a proper map and the Computation Formula (4.1) shows that deg bf (P ) = n, provided that P is canonically top-oriented (see Definition 3.20).
Another important class of bf-admissible triples is given by the Leray-Schauder C 1 -triples. Namely, triples (f, U, q), in which U is a bounded open subset of H, the map f : U → H is a compact vector field, q / ∈ f(∂U ), the restriction f| U is C 1 and has the standard top-orientation (see Definition 3.17). As pointed out in [7], with such a top-orientation, deg bf (f, U, q) is the same as deg LS (f, U, q).
We close this section with the following result regarding Φ 0 -maps in the finite dimensional context. Recall the notions, given in Definition 3.20, of associated and canonical top-orientations. Proof. Taking into account Sard's Lemma, it is a consequence of the definition of the Brouwer degree (see e.g. [19]), Remark 3.8, and the Computation Formula (4.1) of the bf-degree.

Compact and finitely perturbed vector fields
Let H, B, S and r : H\{0} → S be as before. From now onward we shall assume that the dimension of H is at least 3, so that the unit sphere S is simply connected.
Hereafter f = I − k denotes a compact vector field on H. Since f is a compact perturbation of the identity, which is proper on bounded closed subsets of H, f inherits the same property. Therefore, the image under f of a bounded and closed set is as well bounded and closed. In particular, so is f(S).
Recall that the boundary map (of f) is the restriction f| S : S → H of f to S. Moreover, if 0 / ∈ f(S), it makes sense to define the boundary self-map f ∂ : S → S as the composition r • f| S .
By g : H → H we shall denote a finitely perturbed vector field. Namely, a compact vector field with the additional property that the image of the perturbing map h = I − g is contained in a finite dimensional subspace of H. Thus, if H is finite dimensional, any continuous self-map of H is a finitely perturbed vector field. If dim H = ∞, h must send bounded sets into bounded sets, since otherwise g would not be a compact vector field.

Compact vector fields
The following crucial result asserts that the boundary self-map f ∂ , whenever it is defined, is a proper map.
Proof. Observe that the composition of proper maps is a proper map. Nevertheless, f ∂ = r • f| S is not the composition of two proper maps: f| S is proper, but r is not, being defined on its natural domain H\{0}, needed to ensure the composition of r with any boundary map f| S such that 0 / ∈ f(S). However, taking into account that, given f, one has r•f| S = r| f(S) •f| S , the assertion will follow if we prove that the restriction r| f(S) of r to f(S) is a proper map. To check this, recall that f(S) is bounded and closed. Consequently, it is contained in a subset of H\{0} of the type A = [a, b]S, with 0 < a < b < ∞.
Hence, since f(S) is closed, it is enough to show that r is proper on A; and this is true since, given a compact subset C of S, one has (r| A ) −1 (C) = [a, b]C, which is a compact set.
Assume that the compact vector field f = I − k is C 1 . Then the differential df p at any point p ∈ H is given by df p (ṗ) =ṗ−dk p (ṗ), withṗ ∈ T p H = H. Recalling that the Fréchet differential at a point of a compact map is a compact linear operator, we get that df p = I − dk p is a compact linear perturbation of the identity and, consequently, a Φ 0 -operator (see Property 5 of Proposition 2.1). This shows that f is a Fredholm map of index 0. Therefore, since S is a submanifold of H of codimension 1, from Property 2 of Proposition 2.1, we get the following assertion regarding the boundary map f| S . Remark 5.2. If f is of class C 1 , then f| S is a Fredholm map of index −1.
Observe that the differential dr q at any q ∈ H\{0} of the radial retraction r : H\{0} → S, q → q/ q is 1 q Π q ⊥ , where Π q ⊥ is the orthogonal projection of H onto the tangent space T r(q) S = q ⊥ of S at r(q). Thus, dr q is a Fredholm linear operator of index 1 and its kernel is the 1-dimensional subspace Rq of H. Therefore, one gets the following property of the radial retraction.