Some new characterizations of intrinsic transversality in Hilbert spaces

Motivated by a number of research questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. Our dual space results clarify the picture of intrinsic transversality, its variants and the only existing sufficient dual condition for subtransversality, and actually unify them. New primal space characterizations of the intrinsic transversality which is originally a dual space condition lead to new understanding of the property in terms of primal space elements for the first time. As a consequence, the obtained analysis allows us to address a number of research questions asked by the two aforementioned researchers about the intrinsic transversality property in the Hilbert space setting.

Transversality is strictly stronger than subtransversality, in fact, the former property is sufficient for many applications where the latter one is not, for example, in proving linear convergence of the alternating projection method for solving nonconvex feasibility problems [40,41], or in establishing error bounds for the Douglas-Rachford algorithm [16,49] and its modified variants [52]. However, transversality seems to be too restrictive for many applications, and in fact there have been a number of attempts devoted to research for weaker properties but still sufficient for the application of specific interest. Of course, there would not exist any universal transversality-type property that works well for all applications. When formulating necessary optimality conditions for optimization problems in terms of abstract Lagrange multipliers and establishing intersection rules for tangent cones in Banach spaces, Bivas et al. [7] recently introduced a property called tangential transversality, which is a primal space property lying between transversality and subtransversality. When establishing linear convergence criteria of the alternating projection algorithm for solving nonconvex feasibility problems, a series of meaningful transversality-type properties have been introduced and analyzed in the literature: affine-hull transversality [49], inherent transversality [6], separable intersection property [47] and intrinsic transversality [13]. Compared to tangential transversality, the latter ones are dual space properties since they are defined in terms of normal vectors. Compared to the transversality property, all the above transversality-type properties are not dependent on the underlying space, that is, if a property is satisfied in the ambient space X, then so is it in any ambient space containing X. Recall that in the finite dimensional setting, a pair of two closed sets {A, B} is transversal at a common pointx if and only if where N A (x) stands for the limiting normal cone to A atx, see (3) for the definition. This characterization reveals that transversality is a property that involves all the limiting normals to the sets at the reference point. This fundamentally explains why the property is not invariant with respect to the ambient space as well as becomes too restrictive for many applications. Indeed, the hidden idea leading to the introduction of the above dual space transversality-type properties in the context of nonconvex alternating projections is simply based on the observation that not all such normal vectors are relevant for analyzing convergence of the algorithm. Affine-hull transversality is merely transversality but considered only in the affine hull L of the two sets, that is, the pair of translated sets {A−x, B−x} is transversal at 0 in the subspace L−x. As a consequence, the analysis of this property is straightforwardly obtained from that of transversality [49].
The key feature of inherent transversality 1 [6] is the use of restricted normal cones in place of the conventional normal cones in characterization (1) of transversality in Euclidean spaces. As a result, the analysis of this property is reduced to the calculus of the restricted normal vectors as conducted in [6]. The separable intersection property [47,Definition 1] was motivated by nonconvex alternating projections and ultimately designed for this algorithm, and hence it seems to have significant impact only in this context. Intrinsic transversality was also introduced in the context of nonconvex alternating projections in Euclidean spaces [13], it turns out to be an important property itself in variational analysis as demonstrated by Ioffe [22,Section 9.2] and [21] and Kruger [31]. On the one hand, a variety of characterizations of intrinsic transversality in various settings (Euclidean, Hilbert, Asplund, Banach and normed spaces) have been established by a number of researchers [13,21,22,31,35,39,47]. On the other hand, there are still a number of important research questions about this property, for example, the open questions 1-6 asked by Kruger in [31, page 140] or the challenge by Ioffe about primal counterparts of intrinsic transversality [21, page 358]. It is known that for pairs of closed and convex sets in Euclidean spaces, the only existing sufficient dual condition for subtransversality is also a necessary condition, and it is equivalent to intrinsic transversality [31]. Another interesting question is whether the latter two dual properties is equivalent in the nonconvex setting. Motivated mainly by the above research questions, this paper is devoted to investigate further primal and dual characterizations of intrinsic transversality in connection with related properties. Apart from the appeal to address the above research questions, this work was also motivated by the potential for meaningful applications of these transversality-type properties, for example, in establishing convergence criteria for more involved projection algorithms (rather than the alternating projection method) and in formulating calculus rules for relative normal cones (see Definition 4).
The organization of the paper is as follows. New dual space results in the Hilbert space setting are presented in Section 2 with the key estimate obtained in Theorem 1. We show the equivalence between the only existing sufficient dual condition for subtransversality [34,Theorem 1] and the intrinsic transversality property, and provide a refined characterization of the properties, Corollary 1 and Corollary 2. These results significantly clarify the picture of intrinsic transversality, its variants introduced and analyzed in [31] and sufficient dual conditions for subtransversality, and actually unify them. As a consequence, we address an important research question asked by Kruger in [31, question 3, page 140] in the Hilbert space setting. The analysis of intrinsic transversality in finite dimensional spaces is presented in Section 3. The notions of (restricted) relative limiting normal cones [31,Definition 2], which themselves can also be of interest, were introduced and proved to be useful in characterizing transversality-type properties in [31]. We prove that the two cones are equal when restricted on the constraint set C given by (47) only elements in which are of interest for characterizing transversality-type properties, Theorem 2. This new understanding of these normal objects in turn yields insights about intrinsic transversality. As a consequence, we address another important research question asked by Kruger in [31, question 4, page 140] in the Euclidean space setting. New primal space results in the Hilbert space setting are presented in Section 4. We formulate for the first time a primal space characterization of intrinsic transversality, Theorem 3. These results, which substantially open perspective to view intrinsic transversality from primal space elements, were motivated by the research challenge raised by Ioffe in [21, page 358].
Our basic notation is standard; cf. [12,44,50]. The setting throughout the current paper is a Hilbert space X. In order to clearly distinguish elements in the primal space from those in the dual space, we also denote X * the topological dual of X and ·, · the bilinear form defining the pairing between the two spaces. The open unit balls in X and X * are denoted by B and B * , respectively, and B δ (x) (respectively, B δ (x)) stands for the open (respectively, closed) ball with center x and radius δ > 0. The distance from a point x ∈ X to a set Ω ⊂ X is defined by dist(x, Ω) := inf ω∈Ω x − ω , and we use the convention dist(x, Ω) = +∞ when Ω = ∅. The set-valued mapping is the projector on Ω. An element ω ∈ P Ω (x) is called a projection. This exists for any x ∈ X and any closed set Ω ⊂ X. Note that the projector is not, in general, single-valued. If Ω is closed and convex, then P Ω is singleton everywhere. The inverse of the projector, P −1 Ω , is defined by P −1 Ω (ω) := {x ∈ X | ω ∈ P Ω (x) } ∀ω ∈ Ω. The proximal normal cone to Ω at a pointx ∈ Ω is defined by which is a convex cone. Here cone(·) denotes the smallest cone containing the set within the brackets. The Fréchet normal cone to Ω atx is defined by (cf. [24]) which is a nonempty closed convex cone. Here x Ω →x means x →x and x ∈ Ω. The limiting normal cone to Ω atx is defined by In the above definition, the Fréchet normal cone can equivalently be replaced by the proximal one. It holds that N p Ω (x) ⊂ N Ω (x) ⊂ N Ω (x) and that if Ω is closed and dim X < ∞, then N Ω (x) = {0} if and only ifx ∈ bd Ω. By convention, we define N p If Ω is a convex set, then all the above normal cones coincide and reduce to the one in the sense of convex analysis (e.g., [11,Proposition 2.4.4], [24,Proposition 1.19])

Transversality, Subtransversality and Intrinsic Transversality
The following definition recalls possibly the most widely known regularity properties of pairs of sets.
Definition 1 Let {A, B} be a pair of sets in X, andx ∈ A ∩ B.
(i) {A, B} is subtransversal atx if there exist numbers α ∈]0, 1] and δ > 0 such that (ii) {A, B} is transversal atx if there exist numbers α ∈]0, 1] and δ > 0 such that The exact upper bound of all α ∈]0, 1] such that condition (4) or condition (5)  Remark 1 (i) (subtransversality) The subtransversality property was initially studied by Bauschke and Borwein [2] under the name linear regularity as a sufficient condition for linear convergence of the alternating projection algorithm for solving convex feasibility problems in Hilbert spaces. Their results were later extended to the cyclic projection algorithm for solving feasibility problems involving a finite number of convex sets [3]. The term of linear regularity was widely adapted in the community of variational analysis and optimization for several decades, for example, Bakan et al. [1], Bauschke et al. [4,5], Li et al. [42], Ng and Zang [45], Zheng and Ng [55], Kruger and his collaborators [25][26][27][36][37][38]51]. In the survey [18], Ioffe used the property (without a name) as a qualification condition for establishing calculus rules for normal cones and subdifferentials. Ngai and Théra [46] named the property as metric inequality and used it to characterize the Asplund space as well as to establish calculus rules for the limiting Fréchet subdifferential. Penot [48] [43,Section 4] also revealed that the property has been imposed either explicitly or implicitly in all existing linear convergence criteria for nonconvex alternating projections, and hence conjectured that subtransversality is a necessary condition for linear convergence of the algorithm. (ii) (transversality) The origin of the concept of transversality can be traced back to at least the 1970's [15,17] in differential geometry which deals of course with smooth manifolds, where transversality of a pair of smooth manifolds {A, B} at a common pointx can also be characterized by (1) 3 . The property is known as a sufficient condition for the intersection A∩B to be also a smooth manifold aroundx. To the best of our awareness, transversality of pairs (collections) of general sets in normed linear spaces was first investigated by Kruger in a systematic picture of mutual arrangement properties of the sets. The property has been known under quite a number of other names including regularity, strong regularity, property (U R) S , uniform regularity, strong metric inequality [25][26][27]36] and linear regular intersection [40]. Plenty of primal and dual space characterizations of transversality (especially in the Euclidean space setting) as well as its close connections to important concepts in optimization and variational analysis such as weak sharp minima, error bounds, conditions involving primal and dual slopes, metric regularity, (extended) extremal principles and other types of mutual arrangement properties of collections of sets have been established and extended to more general nonlinear settings in a series of papers by Kruger and his collaborators [23,[25][26][27][28][29][30]37,38]. Apart from classical applications of the property, for example, as a sufficient condition for strong duality to hold for convex optimization (Slater's condition) [8,9] or as a constraint qualification condition for establishing calculus rules for the limiting/Mordukhovich normal cones [44, page 265] and coderivatives (in connection with metric regularity, the counterpart of transversality in terms of set-valued mappings) [12,50], important applications have also been emerging in the field of numerical analysis. Lewis et al. [40,41] applied the property to establish the first linear convergence criteria for nonconvex alternating and averaged projections. Transversality was also used to prove linear convergence of the Douglas-Rachford algorithm [16,49] and its relaxations [52]. A practical application of these results is to the phase retrieval problem where transversality is sufficient for linear convergence of alternating projections, the Douglas-Rachford algorithm and actually any convex combinations of the two algorithms [53]. We refer the reader to the recent surveys by Kruger et al. [34,35] for a more comprehensive discussion about the two properties.
A number of dual characterizations of transversality, especially in the Euclidean space setting, have been established [25][26][27]35,36,38,40] and applied, for example, [40,44,49,52]. The situation is very much different for subtransversality. For collections of closed and convex sets, the following dual characterization of subtransversality is due to Kruger.
In the nonconvex setting, the first sufficient dual condition for subtransversality was formulated in [37, Theorem 4.1] following the routine of deducing metric subregularity characterizations for setvalued mappings in [28]. The result was then refined successively in [35,Theorem 4(ii)], [34,Theorem 2] and finally in [31] in the following form.

Proposition 2 [31, combination of Definition 2 and Corollary 2] 5 A pair of closed sets
The inverse implication of Proposition 2 is unknown. Our subsequent analysis particularly shows the negative answer to this question, see Remark 5. Compared to transversality and subtransversality, the intrinsic transversality property below appeared very recently.
simultaneously making an angle strictly less than α with both the proximal normal cones N p B (b) and −N p A (a).
The above property was originally introduced in 2015 by Drusvyatskiy et al. [13] as a sufficient condition for establishing local linear convergence of the alternating projection algorithm for solving nonconvex feasibility problems in Euclidean spaces. As demonstrated by Ioffe [22], Kruger et al. [31,34] and will also be in this paper, intrinsic transversality turns out to be an important property itself in the field of variational anyalysis. Kruger [31] recently extended and investiaged intrinsic transversality in more general underlying spaces.
It is worth noting that the extension from Definition 2 to Definition 3 of intrinsic transversality is nontrivial and the coincidence of the two definitions in the Euclidean space setting was shown in [31, It was proved in [31, Theorem 4] that intrinsic transversality implies the sufficient dual condition of subtransversality provided in Proposition 2, which in turn implies the one stated in Proposition 1. The following quantitative constants [31,34] respectively characterizing the three dual space properties will be convenient for our subsequent discussion and analysis 7 with the convention that the infimum over the empty set equals 1.
In terms of these constants, intrinsic transversality and Propositions 2 and 1 respectively admit more concise descriptions.
The quantitative relationships amongst the five characterization constants defined at Definition 1 and expressions (6)-(7) are as follows.
Proposition 4 [31, Proposition 1] Let {A, B} be a pair of closed sets andx ∈ A ∩ B. 6 The property was defined and investigated in general normed linear spaces. 7 In [31], the restrictions x = a and x = b were also used under the lim inf of (6). We note that they are redundant due to the constraints a ∈ A \ B, b ∈ B \ A and Remark 2 (about notation and terminology) It is clear from Proposition 4(i) that the strict inequality itr w [A, B](x) > 0 characterizes a weaker dual property than intrinsic transversality. That property is indeed called weak intrinsic transversality in [31,34]. This somehow explains why the letter "w" has been used in the notation itr w [A, B](x). Similarly, the strict inequality itr c [A, B](x) > 0 also characterizes some weaker dual property than (weak) intrinsic transversality. Such a property has not been named yet, and it has played an important role in the analysis of transversality-type properties mainly in the convex setting [31]. This somehow explains why the letter "c" has been used in the notation itr c [A, B](x). Since one of the main results of this paper (Corollary 1) reveals that these two constants do coincide with itr[A, B](x) in the Hilbert space setting and, as a result, they characterize the same property -intrinsic transversality, we choose to keep as a minimum number of terminologies as possible in this paper for clarification. It is worth emphasizing that in the general normed linear space setting, such a coincidence remains as a challenging open question and it is natural to treat those properties characterized by the constants itr w [A, B](x) and itr c [A, B](x) independently and as importantly as the intrinsic transversality property, see [31].
We are now ready to formulate one of the main results of this paper. The statement and its proof is rather technical, and its meaningful consequences will be clarified subsequently.
Proof To proceed with the proof, let us suppose that itr c [A, B](x) > 0 since there is nothing to prove in the case itr c [A, B](x) = 0. Let us fix an arbitrary number and prove that itr Choose a number δ ′ ∈ [0, δ/3[ and satisfying Such a number δ ′ exists since We are going to prove itr[A, B](x) ≥ β with the technical constant δ ′ > 0. To begin, let us take any and All we need is to show that We first observe from (16) that We take care of two possibilities concerning the value of x − a, x − b as follows. Equivalently, By the triangle inequality and estimates (19), (18), we get that This implies that Using (20), respectively, we obtain that This combining with (13) yields that Let us define m = a+b 2 and We first check that Indeed, We next check that Indeed, by (22), it holds that Let us define also It is clear that We next check that Let us prove dist(x * 1 ′ , N A (a)) < δ. Indeed, since x * 1 ∈ N A (a) it holds by (25) that has been given by (18): We now establish an upper bound of x−a via three steps as follows.
Conditions (27) and (26) (11) and (12), respectively. It is trivial from the choice of δ ′ at (14) Hence, the estimate (10) Now using the triangle inequality, (40), (41), (42), (14) and (9) successively, we obtain the desired estimate: Due to the technical burden as well as the ease for it discussed in Remark 3, in the remainder of this section, we always make use of the assumption: Corollary 1 (equivalence of dual properties in Hilbert spaces) Let {A, B} be a pair of closed sets andx ∈ A ∩ B. Then it holds that Proof A combination of Theorem 1 and the inequality (43)

yields that itr c [A, B](x) ≤ itr[A, B](x), which together with Proposition 4(i) yields the equalities in (44). ⊓ ⊔
The next result significantly refines Proposition 2 in the Hilbert space setting, which is the weakest sufficient dual condition for subtransversality in the literature.  [31,34]. The question is about the relationship between this property and intrinsic transversality in the general normed space setting. The second equality of (44) clearly shows that the two properties coincide in the Hilbert space setting. Unfortunately, we have not obtained the answer in more general settings.

Corollary 2 (refined sufficient dual condition for subtransversality) A pair of closed sets
In summary, Corollary 1 allows one to unify a number of dual transversality-type properties in the Hilbert space setting including intrinsic transversality, its weaker variant considered in [31], the sufficient dual condition for subtransversality [34,35,38] and the dual characterization of subtransversality with convexity [31]. In our opinion, this significantly clarifies the picture of these important dual space properties.

Intrinsic Transversality in Finite Dimensional Spaces
We first recall definitions about relative limiting normals which are motivated by the compactness of the unit ball in finite dimensional spaces as well as the fact that not all normal vectors are always involved for characterizing transversality-type properties. These notions were shown to be useful for analyzing the intrinsic transversality property and its variants, see [31, page 123] for a more thorough discussion.

Definition 4 [31, Definition 2] Let
with the convention that 0 0 = 1. The collections of all pairs of relative limiting normals to {A, B} at x will be denoted by The collections of all pairs of restricted relative limiting normals to {A, B} atx will be denoted by N Thanks to the compactness of the sets under the following minima in finite dimensional spaces, making use of the notions in Definition 4 leads to alternative and more concise representations for itr[A, B](x) and itr c [A, B](x) compared to (6) and (7), respectively: with the convention that the minimum over the empty set equals 1.
We have shown in Corollary 1 that the two constants are either both greater than 1/ √ 2 or equal in the Hilbert space setting. How about the relationship between the two cones under the minima on the right-hand-side of (45) and (46)? It has only been known from [31]  To begin, let us define the constraint set below:

Remark 8
The set C is consistently related to the assumption (43) we imposed in Section 2. More specifically, condition (43) holds true if and only if Under this condition, one can further refine the two expressions (45) and (46): and, as a consequence, only pairs of vectors in C are needed for calculating the two quantitative constants.
In view of (48) and (49) and Corollary 1, a number of characterizations of intrinsic transversality can be recast in the next proposition, which slightly extends the list in [31, Theorems 5 and 6].
Proposition 6 Let X be finite dimensional, A, B ⊂ X be closed andx ∈ A∩B. The following conditions are equivalent: If, in addition, the sets A, B are convex, then the following item can be added to the above list.

Remark 9
It is also possible to establish an expression for itr w [A, B](x) analogous to (48) and (49)  However, we will show later on in Theorem 2 that the latter two cones are indeed equal when restricted in the cone C that is the case of our main interest (see Remark 8). Hence, we choose not to give details about this task for simplicity in terms of presentation.
We now reveal a deeper relationship between N A,B (x) and N c A,B (x). This result complements Corollary 1 in the Euclidean space setting and further clarifies the characterization of intrinsic transversality in terms of (restricted) relative limiting normals. Apart from the latter application, the next theorem was also inspired by the importance of the cones themselves, see [31, page 123].
Theorem 2 Let {A, B} be a pair of closed sets in a Euclidean space andx ∈ A ∩ B. Then Proof It is known by [31,Proposition 2(ii) Since (x * 1 , x * 2 ) ∈ C, it holds that x * 1 , x * 2 ≤ 0, equivalently, To complete the proof, it suffices to prove that (x * 1 , x * 2 ) ∈ N c A,B (x). For each k = 1, 2, . . ., let us define: All we need is to verify the following four conditions: Condition (i): this follows from (52) since for each k = 1, 2, . . ., we have that Condition (iv): from (53), we have that , whenever condition (ii) has been verified, we have that Condition (ii): since x k →x, a k →x and b k →x, it holds by (52) that In the remainder of the proof, we show that x * 1k ′ → x * 1 while the condition x * 2k ′ → x * 2 is obtained in a similar manner. Since x * 1k → x * 1 , we need to show that x * 1k ′ − x * 1k → 0. Note that by (53) it holds that Note also that due to (50), In view of (55) and (56), to obtain x * 1k ′ − x * 1k → 0, it suffices to prove that To proceed, let us take any number ε > 0 which can be arbitrarily small and show the existence of an natural N ∈ N such that Choose a number ε ′ > 0 and satisfying Such a number ε ′ exists since ε > 0 and lim t↓0 2 2t−t 2 4−6t+3t 2 = 0. By the convergence conditions in (50), there exists a natural number N ∈ N such that ∀k ≥ N , The estimates in (60) amount to In order to prove (57), we first note that Indeed, by (52), it holds that Second, we show that ∀k ≥ N , Note from (51) and (61) that This combining with (59) yields that From (52), (64) and (65) we have that By (54) and (62) we get that This together with (59) and (66) yields Hence By (54) and (62) we get that which together with (59) and (68) yields that Equivalently, Combining (67) and (69) and noting that 2ε Indeed, if x k − a k ≤ x k − b k , then the use of (68) and (69) yields (70): Otherwise, i.e., x k − a k ≥ x k − b k , then the use of (54), (66) and (63) successively implies that which also yields (70) since 4+2ε ′ +ε ′2 2ε ′ +ε ′2 > 4−6ε ′ +3ε ′2 2ε ′ −ε ′2 . Hence (70) has been proved. Fourth, we show that ∀k ≥ N Indeed, Hence (71) has been proved. Finally, a combination of (70), (71) and (58) yields that which is (57) and hence the proof is complete.

Remark 10
In the Euclidean space setting, thanks to (48) and (49), Corollary 1 can easily be deduced from Theorem 2. But the inverse implication is not trivial since the minimal values at (48) and (49) being equal does not tell much about the relationship between the two feasibility sets there.

Primal Space Characterizations of Intrinsic Transversality
In the Hilbert space setting, it can be deduced from Proposition 4(iii) and Corollary 1 that for pairs of closed and convex sets, intrinsic transversality is equivalent to subtransversality which is a primal space property. The situation for pairs of nonconvex sets has not been known and there is an interest to research for primal space counterparts of intrinsic transversality in this setting. This research question was raised by Ioffe [22, page 358]. Our agenda in this section is to present material sufficient for formulating a primal characterization of intrinsic transversality in the Hilbert space setting.
In the sequel, we always assume that the Cartesian product space X×X is endowed with the maximum norm and accordingly define the distance between two subsets of as follows: for any P, Q ⊂ X × X, dist (P, Q) := inf (p1,p2)∈P,(q1,q2)∈Q with the convention that the infimum over the empty set equals infinity. For convenience, for a subset Ω ⊂ X, we use the following notation: [ Note that [Ω] 2 is different from (smaller than) the Cartesian product set Ω × Ω. We frequently use the distance (72) involving a Cartesian product set and a set of form (73): We formulate several technical results which are essential for proving the key estimates in this section.
Proposition 7 [10, Corollary 6.3] 10 Let A, B be a pair of closed sets in X,x ∈ A ∩ B, u, v ∈ X and numbers ρ, ε > 0. Suppose that Then, for any numbers λ ≥ ε + ρ and τ ∈ 0, λ−ε λ+ε there exist pointsã Condition (78) plays an important role in our analysis. It relates the dual space elements The next lemma slightly modifies Proposition 7 in such a way that the imposed condition about the common point of the sets can be relaxed.

⊓ ⊔
We now state and prove the key estimates for our analysis.
Theorem 3 (key estimates) Let {A, B} be a pair of closed subsets of X andx ∈ A ∩ B and consider the following statements.
which tends to 1 as ε ↓ 0. Thus, Due to the Cauchy-Schwarz inequality and x * 1 + x * 2 = 1, the above convergence happens if and only if In view of (105), (106) and the five observations above, by letting ε ↓ 0 and comparing the definition Such a number ε exists since 2t(α + 1/ √ t) → 0 as t ↓ 0. By the construction (7) Then by Lemma 2 with noting that α 1 + 2ε(α + 1/ √ ε) < α by (108) The first estimate of (101) shows that the primal space property (ii) of Theorem 3 is a necessary condition for intrinsic transversality, while the second one shows that the property is a sufficient condition for the property characterized by itr c [A, B](x). A combination of Theorem 3 and Theorem 1 bridges the gap between the these properties in the Hilbert space setting. As a result, we establishes the primal space characterization of intrinsic transversality for the first time.
Proof The first statement (i) easily follows from Remark 3 and the first estimate of (ii), while the second one (ii) directly follows from Corollary 1 and ii.
An important implication of Corollary 3 is that the property (ii) of Theorem 3 is indeed a primal space characteriztion of intrinsic transversality as desired. Moreover, the equality itr[A, B](x) = itr p [A, B](x) at (111) completes the quantitative relationship between the two primal and dual space counterparts in the case of most interest.