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On the notion of scalar product for finite-dimensional diffeological vector spaces

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to circumnavigate this issue, by introducing a notion of pseudo-metric, which, said informally, is the least-degenerate symmetric bilinear form on a given space. We apply this notion to make some observation on subspaces which split off as smooth direct summands (providing examples which illustrate that not all subspaces do), and then to show that the diffeological dual of a finite-dimensional diffeological vector space always has the standard diffeology and in particular, any pseudo-metric on the initial space induces, in the obvious way, a smooth scalar product on the dual.


Introduction
One of the first surprising findings which one makes when encountering diffeology for the first time is this inequality: L ∞ (V, W ) < L(V, W ), which says that the space of all (diffeologically) smooth linear maps between two diffeological vector spaces, V and W , can be strictly smaller than that of all maps that are simply linear. 1 Stemming from that, an even more surprising situation presents itself: a (finitedimensional) diffeological vector space that does not admit a smooth scalar product. This is a known fact (see [1], p. 74, Ex. 70), and easily established at that, yet, it is still surprising, in and of itself, and also for how easily this can be illustrated, using the presence of just one non-differentiable (in the usual sense) plot.
Thus, as mentioned in the above-given reference, any finite-dimensional diffeological vector space that admits a smooth scalar product is necessarily the usual R n , with its usual smooth structure (the diffeology that consists of all usual smooth maps). The choices at this point are, to abandon the whole affair (meaning to concentrate on infinite-dimensional spaces, where the similar situation does not seem to occur), to consider a kind of pseudo-metrics (meaning the sort of least degenerate symmetric bilinear form that exists on a given space), or, finally, to re-define scalar product as ones taking values in R endowed, not with the standard diffeology, but with the piecewise-smooth diffeology. We concentrate on the second of the above approaches, and it does lead to a few interesting conclusions, mainly regarding the diffeological dual and the fact that a pseudo-metric induces an isomorphism of it with a specific subspace, which is a smooth summand. Furthermore, it gives rise to a smooth scalar product on the dual, reflecting as much as possible of the usual duality for the standard vector spaces.
Finally, a disclaimer: a lot of what is written here might be of a kind of implicit knowledge for people working in the area. Part of the aim of this paper is to collect these facts in one place, and to make explicit what is implicit elsewhere. 2 Acknowledgments Some of the best things in life happen by chance: a chance encounter, a chance phrase, a chance look... This sounds trivial, but what is surprising is that I found this true of professional life as well; some of my most productive moments came about by chance. The very existence of this work is due to chance, and for it, I would like to thank the people who, incidentally, are also responsible for some of the best moments in the last couple of years (all of them, of course, due to chance). These people are: Prof. Riccardo Zucchi, Prof. Patrick Iglesias-Zemmour, Prof. Danilo De Rossi. I also would like to thank all the participants of the workshop "On Diffeology etc." (Aix en Provence, June [24][25][26]2015) where I had my first opportunity to discuss the reasoning that is at the origin of this paper.

Diffeology and diffeological vector spaces
We briefly recall here the main definitions regarding diffeological spaces and diffeological vector spaces.
Diffeological spaces and smooth maps We first recalling the notion of a diffeological space and that of a smooth map between such spaces.
where X is a set and D X is a specified collection of maps U → X (called plots) for each open set U in R n and for each n ∈ N, such that for all open subsets U ⊆ R n and V ⊆ R m the following three conditions are satisfied: 1. (The covering condition) Every constant map U → X is a plot; 2. (The smooth compatibility condition) If U → X is a plot and V → U is a smooth map (in the usual sense) then the composition V → U → X is also a plot; 3. (The sheaf condition) If U = ∪ i U i is an open cover and U → X is a set map such that each restriction U i → X is a plot then the entire map U → X is a plot as well.
Usually, instead of (X, D X ) one writes simply X to denote a diffeological space.
Definition 1.2. ( [5]) Let X and Y be two diffeological spaces, and let f : X → Y be a set map. The map f is said to be smooth if for every plot p : U → X of X the composition f • p is a plot of Y .
The standard examples of diffeological spaces are smooth manifolds, with diffeology consisting of all usual smooth maps R k ⊃ U → M , for all k ∈ N and for all domains U in R k . A less standard example is, for instance, any Euclidean space R n with the so-called wire diffeology, namely, the diffeology generated (see the next paragraph) by the set C ∞ (R, R n ).
Generated diffeology Let X be a set, and let A = {U i → X} i∈I be a set of maps with values in X. The diffeology generated by A is the smallest, with respect to inclusion, diffeology on X that contains A. It consists of all maps f : V → X such that there exists an open cover {V j } of V such that f restricted to each V j factors through some element U i → X in A via a smooth map V j → U i . Note that this construction illustrates the abundance of diffeologies on a given set: we can build one starting from any set of maps; in particular, given any map p : U → X, there is (usually more than one) diffeology containing it.
Fine diffeology and coarse diffeology Given a set X, the set of all possible diffeologies on X is partially ordered by inclusion, namely, a diffeology D on X is said to be finer than another diffeology D ′ if D ⊂ D ′ (whereas D ′ is said to be coarser than D). Among all diffeologies, there is the finest one, which turns out to be the natural discrete diffeology and which consists of all locally constant maps U → X; and there is also the coarsest one, which consists of all possible maps U → X, for all U ⊆ R n and for all n ∈ N. It is called the coarse diffeology.
Subset diffeology Let X be a diffeological space, and let Y ⊂ X be a subset of it. The subset diffeology on Y is the coarsest diffeology on Y such that the inclusion map Y ֒→ X is smooth. What this means specifically, is that p : R k ⊃ U → Y is a plot for the subset diffeology on Y if and only if its composition with the inclusion map is a plot of X.
Products and sums of diffeological spaces Let {X i } i∈I be a collection of diffeological spaces, where I is a set of indices. The product diffeology D on the product i∈I X i is the coarsest diffeology such that for each index i ∈ I the natural projection π i : i∈I X i → X i is smooth. Note, in particular, that for the product of two spaces X 1 × X 2 every plot is locally of form (p 1 , p 2 ), where p i is a plot of X i for i = 1, 2.
If we now consider the disjoint union i∈I X i of the spaces X i , this can be endowed with the sum diffeology, which by definition is the finest diffeology such that all the natural injections X i ֒→ i∈I X i are smooth. The plots for this diffeology are those maps that are locally plots of one of the components of the sum.
Diffeological vector space Let V be a vector space over R. The vector space diffeology on V is any diffeology of V such that the addition and the scalar multiplication are smooth, that is, where V × V and R × V are equipped with the product diffeology. A diffeological vector space over R is any vector space V over R equipped with a vector space diffeology.
Fine diffeology on vector spaces The fine diffeology on a vector space R is the finest vector space diffeology on it; endowed with such, V is called a fine vector space. As an example, R n with the standard diffeology is a fine vector space.
The fine diffeology admits a more or less explicit description of the following form: its plots are maps f : U → V such that for all x 0 ∈ U there exist an open neighbourhood U 0 of x 0 , a family of smooth maps λ α : U 0 → R, and a family of vectors v α ∈ U 0 , both indexed by the same finite set of indices A, such that A finite family (λ α , v α ) α∈A , with λ ∈ C ∞ (U 0 , R) and v α ∈ V , defined on some domain U 0 and satisfying the condition just stated, is called a local family for the plot f .
Fine vector spaces possess the following property ([1], 3.9), which in general not true: if V is a fine diffeological vector space, and W is any other diffeological vector space, then every linear map V → W is smooth, i.e., L ∞ (V, W ) = L(V, W ).
Smooth scalar products The existing notion of a Euclidean diffeological vector space does not differ much from the usual notion of the Euclidean vector space. A diffeological space V is Euclidean if it is endowed with a scalar product that is smooth with respect to the diffeology of V and the standard diffeology of R; that is, if there is a fixed map , : V × V → R that has the usual properties of bilinearity, symmetricity, and definite-positiveness and that is smooth with respect to the diffeological product structure on V × V and the standard diffeology on R. As has already been mentioned in the Introduction, for finite-dimensional spaces this implies that the space in question is just the usual R n , for appropriate n (see [1]). For reasons of completeness we recall the precise statement and give a detailed proof of this fact. 3 Proposition 1.3. Let V be R n endowed with a vector space diffeology D such that there exists a smooth scalar product. Then every plot p of D is a smooth map in the usual sense.
Proof. Let A an n × n non-degenerate symmetric matrix such that the associated bilinear form on V is smooth with respect to the diffeology D, and let {v 1 , . . . , v n } be its eigenvector basis. Let λ i be the eigenvalue relative to the eigenvector v i .
Let p : U → V be a plot of D; we wish to show that it is smooth as a map U → R n . Recall that ·|· A being smooth implies that for any two plots p 1 , this is of course a plot of D. Set p 1 = p and p 2 = c i ; then the above composition map writes as λ i p(x)|v i , where ·|· is the canonical scalar product on R n .
Since A is non-degenerate, all λ i are non-zero; this implies that each function p(x)|v i is a smooth map. And since v 1 , . . . , v n form a basis of V , this implies that for any v ∈ V the function p(x)|v is a smooth one. In particular, this is true for any e j in the canonical basis of R n ; and in the case v = e j the scalar product p(x)|e j is just the j-th component of p(x). Thus, we obtain that all the components of p are smooth functions, therefore p is a smooth map.

The degeneracy of smooth forms on non-standard spaces
As mentioned in the introduction, pretty much all finite-dimensional diffeological vector spaces do not have smooth non-degenerate bilinear forms; there is only one for each dimension that does, and that is the standard one. For exposition purposes, we start with a detailed illustration of how this happens, via an example (this part is however easily deduced from the proof of Proposition 1.3 above). We then consider the degree of degeneracy of smooth bilinear forms on a given vector space (what does it mean, for a given V , to be the least-degenerate bilinear form on it?), going on to the question of subspaces which do, or do not, split off as smooth direct summands, and finally considering the diffeological dual.

Pseudo-metrics
A pseudo-metric is, roughly speaking, the best possible substitute for the notion of a smooth scalar product in the case of a finite-dimensional vector space whose diffeology is not the standard one.
The first example The following example, already considered in [2], is presented for illustrative purposes.
Example 2.1. Let V = R n , and let v 0 ∈ V be any non-zero vector. Let p : R → V be defined as p(x) = |x|v 0 ; let D be any vector space diffeology on V that contains p as a plot. 4 Suppose that A is a symmetric n × n matrix, and assume that the bilinear form v|w A = v t Aw associated to A is smooth with respect to D and the standard diffeology on R. We claim that A is degenerate.
Indeed, v|w A being smooth implies, in particular, that for any two plots p 1 , p 2 : R → V of V the composition map ·|· A • (p 1 , p 2 ) : R → R is smooth in the usual sense; this map acts as R ∋ x → (p 1 (x)) t Ap 2 (x). Let w ∈ V be an arbitrary vector; denote by c w : R → V the constant map that sends everything to w, c w (x) = w for all x ∈ R. Such a map is a plot for any diffeology on V . But then ( ·|· A • (p, c w ))(x) = |x|v t 0 Aw; the only way for this to be smooth is to have v t 0 Aw = 0, and since there was no assumption on w, this implies that v 0 |· A is identically zero on the whole of V , i.e., that A is degenerate. In other words, V does not admit a smooth scalar product.
Note that in the above example we could have taken that is not differentiable in at least one point; this suggests that there are numerous diffeological vector spaces that do not admit diffeologically smooth scalar products. In fact, Proposition 1.3 shows that the phenomenon is much more general: in order to have a smooth scalar product, we must ensure that all plots are smooth maps.
The signature of a smooth bilinear form Suppose that we have a finite-dimensional diffeological vector space V ; let A be a symmetric n × n matrix (with n = dim V ) such that the associated bilinear form ·|· A on V is smooth. Let (λ + , λ − , λ 0 ) be the signature of this form; recall that V * stands for the diffeological dual of V (see [6] and [7]), i.e. the set of all smooth linear functionals on V . 4 Such diffeology does certainly exist; for instance, the coarse diffeology would do.
Proof. Choose a basis {v 1 , . . . , v n } of eigenvectors of A such that the last λ 0 vectors are those relative to the eigenvalue 0. For i = 1, . . . , n − λ 0 let v i ∈ V * be the dual function, that is, v i (w) = v i |w A (it is obviously smooth since ·|· A is smooth, so is an element of V * ). It remains to notice that by standard reasoning the elements v 1 , . . . , v n−λ0 are linearly independent (they belong to non-zero eigenvalues), so n − λ 0 dim(V * ), hence the conclusion.
Notice that we can always find a smooth bilinear (symmetric) form for which we have λ 0 = n − dim(V * ). It suffices to take any basis {f 1 , . . . , f k } of V * (with k = dim(V * )) and consider i f i ⊗ f i ; this is obviously a symmetric bilinear form on V , and it being smooth follows from Theorem 2.3.5 of [6]. By an obvious argument, we can see that all the eigenvalues are non-negative; indeed, 0. Finally, the multiplicity of the eigenvalue 0 is precisely n − k, since f 1 , . . . , f k form a basis of V * .
Thus, the following definition makes sense.
A diffeological pseudometric on V is a smooth bilinear symmetric form on V such that the eigenvalue 0 has multiplicity n − dim(V * ), and all the other eigenvalues are positive.
A pseudo-metric is the best notion of a smooth 5 metric that can exist for an arbitrary finitedimensional diffeological vector space; note that if a given space is standard, i.e. is of the only type which admits a smooth metric, then obviously a pseudo-metric is a usual Euclidean metric. Of course, one can look for true metrics which are not smooth but are at least piecewise-differentiable, or simply continuous, something that we will do in the section that follows; meanwhile, we explore whatever applications pseudo-metrics might have.
Remark 2.4. While it is more usual to use scalar products for various constructions of (multi)linear algebra, we do note that a number of them hold in part for symmetric bilinear forms. A specific example would be a Clifford algebra, which is defined for a vector space endowed with a symmetric bilinear form that a priori does not have to be a scalar product (an extension of this notion into the diffeological context was considered in [3]). Obviously, if it is not then one cannot speak of unitary action on the exterior algebra, for example; yet, various constructions hold.

Smooth direct sums and pseudo-metrics
One easily observed fact that distinguishes diffeological vector spaces from their usual linear counterpart is that, in the diffeological case, there are two types of direct sum: one that is smooth (in the sense described just below) and the other that is not.
Smooth splittings as direct sums In general, if we have a diffeological vector space V (of finite dimension) that, as a usual vector space, decomposes into a direct sum V = V 1 ⊕ V 2 , then a priori the following two diffeologies are different: • the given diffeology of V , and • the vector space sum diffeology on V 1 ⊗ V 2 obtained from the subset diffeologies on V 1 and V 2 . 6 Example 2.5. Let us consider an example where the two might actually be different. Take V = R 3 endowed with the vector space diffeology that is generated by the map p : R → V acting by p(x) = |x|(e 2 + e 3 ). Set V 1 = Span(e 1 , e 2 ) and V 2 = Span(e 3 ). Observe that the subset diffeology on each of these subspaces is just the standard diffeology of R 2 and R respectively. 7 This means that the vector space sum 5 Which means a bilinear form that takes values in R with standard diffeology and is smooth with respect to the latter. 6 The vector space sum diffeology on a direct sum of diffeological vector spaces is the finest vector space diffeology that contains the sum diffeology; it is formed of all maps that locally are (formal) sums of plots of the summands. 7 Let us prove this claim. Let q : U → V 1 be a plot for the subset diffeology on V 1 ; let q i : U → R be the projection on Span(e i ) V 1 for i = 1, 2. It suffices to show that q i must necessarily be a smooth map (in the usual sense). Note that by definition of the subset diffeology Q(u) = (q 1 (u), q 2 (u), 0) must be a plot for the diffeology of V . By definition of the latter diffeology in the neighbourhood of zero it writes as Q(u) where f 1 , f 2 , f 3 , F are the usual smooth maps U → R; therefore up to smooth summands Q(u) writes as (0, |F (u)|, |F (u)|), from which we conclude that |F (u)| ≡ 0 in the neighbourhood of u = 0. This implies that q 1 , q 2 are the usual smooth maps, and the claim is proved for V 1 ; the reasoning is analogous in the case of V 2 .
diffeology on their direct sum is again the standard one, which the diffeology on V is not; indeed, the projection on the second coordinate is smooth (in the usual sense) for the standard diffeology, which does not happen for the diffeology of V .
Standard subspaces that split off as direct summands The example given in the previous paragraph easily leads to some more general observations, such as the following.
Lemma 2.6. Let V be a finite-dimensional diffeological vector space, let V 1 V be its subspace such that its respective subset diffeologies is standard, and let V = V 1 ⊕ V 2 be its decomposition into a direct sum for some subspace V 2 V . This decomposition is smooth if and only if there exists a basis {v 1 , . . . , v k } of V 1 such that the projection 8 on each vector v i is a smooth linear functional on V .
Proof. The part "only if" of the statement follows from the definition of the (vector space) direct sum diffeology. Indeed, suppose that the diffeology on V coincides with the direct sum diffeology formed from the subset diffeologies on V 1 and V 2 ; recall that the plots of standard diffeology are of form i f i (x)v i , for some smooth functions f i and a basis {v 1 , . . . , v k }.
Let π i : V → R be the projection on the vector v i . Recall that each plot p : U → V of V locally writes as p = p 1 ⊕ p 2 , where (in particular) p 1 is a plot of V 1 , hence (π i • p)(u) = π i (p 1 (u)). The latter is a smooth map U → R, since the diffeology of V 1 is standard; therefore π i is smooth as a map V → R.
Let us now prove the "if" part. Let, again, π i : V → R be the projection on the vector v i ; by assumption, this is a smooth map, i.e., for any plot p : U → V of V the composition π i • p : U → R is smooth in the usual sense. Now, in the usual linear algebra sense p does write as p = p 1 ⊕ p 2 , where p 1 : U → V 1 and p 2 : U → V 2 ; we need to show that each of these p i is a plot of V i .
Let us first prove this for p 1 . Note that we have p 1 (u) = π 1 (u)v 1 + . . . + π k (u)v k by definitions of the maps involved; since all π i 's are usual smooth maps by assumption, this is of course a plot of V 1 . Note also that the formula just given allows to regard p 1 also as a plot of V . Then p 2 = p − p 1 (where p 2 is also considered as a map in V in the obvious sense), and since the diffeology of V is a vector space diffeology, p 2 is a plot of V . By definition of the subset diffeology, as a map in V 2 it is a plot of it. This implies that the diffeology of V is indeed the (vector space) sum diffeology.
We have already given an example for which this claim produces a non-trivial result. It also can be generalized as follows (the statement is rather complicated and probably not very useful in itself; it does underline what truly matters in the above proof).
Observation 2.7. Let V be a finite-dimensional diffeological vector space, and let V 1 , V 2 V be its usual vector space subspaces such that V = V 1 ⊕ V 2 . This decomposition is smooth if and only if there exist bases {v 1 , . . . , v k } of V 1 and {v k+1 , . . . , v k+l } of V 2 satisfying the following condition: if D i for i = 1, 2 is the finest diffeology on R that contains all projections V i → R on the vectors of the corresponding basis, then either D 1 ⊂ D 2 or D 2 ⊂ D 1 .
The maximal standard subspace relative to a pseudo-metric As follows from the above discussion, a pseudo-metric exists on any diffeological vector space (and we cannot do any better). On the other hand, given a pseudo-metric ·|· A on a fixed diffeological vector space (V, D) of finite dimension, the subspace V 0 of V spanned by all the eigenvectors belonging to the positive eigenvalues of ·|· A and considered with the subset diffeology D 0 induced by D is endowed, via the restriction of ·|· A , with a true metric; it follows then from [1], Ex. 70, p.74 that (V 0 , D 0 ) is the standard Euclidean space of dimension n − λ 0 = n − dim(V * ).
Lemma 2.8. The subspace V 0 splits off as a smooth direct summand.
Proof. Denote by V 1 the subspace generated by all the eigenvectors relative to the eigenvalue 0. Obviously, V splits as the direct sum V = V 0 ⊕ V 1 in the usual sense; we need to check that this decomposition is smooth, that is, that the diffeology on V coincides with the direct sum diffeology on V 0 ⊕ V 1 . Now, by the lemma in the preceding paragraph, it suffices to check that the projection on each eigenvector in some basis of V 0 is a smooth linear functional on the whole of V .
The matrix A being symmetric, there exists an orthonormal (with respect to the usual scalar product on R n underlying V ) basis {v 1 , . . . , v n } of V composed of eigenvectors of A, where we can obviously assume that the eigenvectors relative to all the positive eigenvalues (and so forming a basis of V 0 ) are the first k vectors, v 1 , . . . , v k . Let v i be one of these, relative to the eigenvalue λ i , let π : V → R be the corresponding projection, and let p : U → V be any plot of V ; we need to show that π • p : U → R is smooth in the usual sense. Write p(x) = n j=1 p j (x)v j ; recall that p i (x) = π • p. Since A defines a smooth bilinear form on V , and all the constant maps are plots, the assignment x → v i |p(x) A defines a map U → R which is smooth in the usual sense. However, we have v|p( where ||v i || is the usual Euclidean norm of v i . Since neither it nor λ i are zero, we conclude that π • p is smooth, as wanted.
Furthermore, the following is true.
Proposition 2.9. The subspace V 0 is a maximal subspace of V with the subset diffeology that is standard and which splits off as a smooth direct summand.
Proof. Suppose that V 1 > V 0 is a bigger subspace of V such that its subset diffeology is standard. Let v 1 , . . . , v k be all the (linearly independent) eigenvectors of A relative to the positive eigenvalues; by definition of V 0 , they form a basis of it. Let v ∈ V 1 \ V 0 be a vector linearly independent from v 1 , . . . , v k .
Set ϕ(w) = w|v (the canonical scalar product on V ); let us show that ϕ is smooth as a linear functional on V . Indeed, if p : U → V is a plot of V then ϕ(p(x))v is the usual orthogonal projection of p(x) on the line generated by v. Since V 0 splits off as a smooth direct summand, by Lemma ?? the projection on v is smooth, therefore ϕ belongs to the diffeological dual V * of V . Recall now that dim(V * ) = k, and we can obtain its basis f 1 , . . . , f k by setting f i (w) = w|v i A for each i = 1, . . . , k. Then ϕ writes as ϕ = α 1 f 1 + . . . + α k f k for some α 1 , . . . , α k not all of which are zero. Since v i 's are eigenvectors, we then get w|v = α 1 λ 1 w|v 1 + . . . + α k λ k w|v k ; since this is true for an arbitrary w and the scalar product is the canonical one, this implies that v = α 1 λ 1 v 1 + . . . + α k λ k v k . Since by assumption all λ 1 , . . . , λ k are positive, and v is linearly independent from v 1 , . . . , v k , we get a contradiction.
In other words, a pseudo-metric on V allows to extract, in a sense, from V its biggest standard part, on which the diffeology includes essentially the usual smooth maps. Let us be more precise.
Lemma 2.10. Let V = W 0 ⊕ W 1 be a smooth decomposition such that dim(W 0 ) dim(V * ), and the subset diffeology on W 0 is standard. Then there exists a pseudo-metric ·|· A on V such that W 0 is the space generated by the eigenvectors of A relative to the positive eigenvalues, and W 1 is the space generated by the eigenvectors relative to the eigenvalue 0.
Proof. Since the decomposition is smooth, each plot q of V is locally of form q 0 +q 1 for some plots q 0 , q 1 of W 0 , W 1 respectively. Now, the subset diffeology on W 0 being standard, there exists a basis {w 1 , . . . , w k } of W 0 such that each plot q 0 is (locally) of form q 0 (u) = k i=1 q i 0 (u)w i . Take some basis {w k+1 , . . . , w n } of W 1 ; then {w 1 , . . . , w n } is of course a basis of V , and it suffices to define B as (B) ii = 1 for i = 1, . . . , k and (B) ij = 0 otherwise.
To summarize, all subspaces of V with standard sub-diffeology that in addition split off as smooth direct summands are associated to some pseudo-metric, provided they have a sufficiently large dimension. We can actually show more: there is only one of them.
Proposition 2.11. The subspace V 0 is an invariant of the diffeological space V , i.e., it does not depend on the choice of a pseudo-metric ·|· A .
Proof. Let W 0 = V 0 be another subspace of V of dimension k = dim(V * ) such that the subset diffeology on W 0 is standard and W 0 splits off as a smooth summand, V = W 0 ⊕ W 1 . Let w 0 ∈ W 0 \ W 1 ; it writes uniquely as w 0 = v 0 + v 1 . Note that by Lemma 2.6 the projections on both w 0 and v 0 are smooth linear functionals; therefore so is the projection on v 1 . That would imply that Span(V 0 , v 1 ) has standard diffeology and splits off as a smooth summand; since its dimension is strictly greater than dim(V 0 ) = dim(V * ), this is a contradiction.

The metric on V * induced by a pseudo-metric on V
Here we consider a finite-dimensional diffeological vector space V (we assume that its underlying vector space is identified with R n ), endowed with a pseudo-metric ·|· A given by a symmetric matrix A. As for usual vector spaces, the pseudo-metric induces a symmetric bilinear form on the diffeological dual V * of V ; in this section we deal with this induced form.
The functional diffeology on V * Recall [7] that in general, the diffeological dual V * of V is not even isomorphic to V , so not diffeomorphic to it; this occurs already in the finite-dimensional case (in addition to [7], see Example 3.1 in [2]). Thus, it might well occur that while the diffeology on V is not standard, the corresponding functional diffeology on its dual is. The following statement makes it precise.
Theorem 2.12. Let V be a finite-dimensional diffeological vector space, and let V * be its diffeological dual. Then the functional diffeology on V * is standard ( i.e., V * is diffeomorphic to some R k with standard diffeology, for the appropriate k).
Proof. Let us choose any pseudo-metric ·|· A on V (as we have already noted, we can always find one). Let {v 1 , . . . , v n } be an orthonormal basis of V such that the eigenvectors relative to the positive eigenvalues are the first k vectors; by an above lemma, this means that the dimension of V * is equal to k. Denote by λ i the eigenvalue relative to the eigenvector v i ; note that for each i = 1, . . . , k the map f i : V → R given by f i (w) = 1 λi w|v i A is a smooth linear functional on V , and so an element of V * . Finally, observe that each f i coincides with v i , the usual dual of the vector v i , from which we conclude that {f 1 , . . . , f k } is a basis of the diffeological dual V * .
To show that the diffeology of V * is the standard one, it is sufficient to show that any plot p : U → V * of it locally writes as p(u) = p 1 (u)f 1 + . . . + p k (u)f k for some smooth (in the usual sense) functions p i : U → R. That it does write so, for some functions p i , obviously follows from the fact that {f 1 , . . . , f k } is a basis, so we only need to show that each p i is a smooth function.
Recall ([1], Section 1.57) that, the diffeology of V * being functional, p is a plot of it if and only if the function ϕ p : U × V → R given by ϕ p (u, v) = p(u)(v) is smooth (for the product diffeology). By definition of the latter, in particular, the following map is ordinarily smooth: ϕ p • (Id U , c vi ) : U → R, where Id U is the identity map on U and c vi is the constant map that sends each u ∈ U in v i . But (ϕ p • (Id U , c vi ))(u) = p(u)(v i ) = p i (u), since the vectors v i form an orthonormal basis. This allows us to conclude that each p i is indeed an ordinary smooth map U → R, which, as already mentioned, implies that the diffeology of V * is the standard one.
The subspace V 0 and the diffeological dual The proofs of Lemma 2.8 and of Theorem 2.12 suggest that each pseudo-metric on V gives a natural identification between the corresponding V 0 and V * (that the two spaces are a priori diffeomorphic is now obvious, since they are both standard spaces of the same dimension). More precisely, the following is true.
Theorem 2.13. Let V be a finite-dimensional diffeological vector space, and let ·|· A be a pseudo-metric on V . The restriction on the subspace V 0 of the induced map Ψ : V → V * ( i.e., the map given by Proof. As has been already observed in the proof of Theorem 2.12, the subspace V 0 admits an orthonormal (with respect to the canonical scalar product on R n underlying V ) basis {v 1 , . . . , v k } composed of eigenvectors of A; furthermore, if f i = Ψ(v i ) for i = 1, . . . , k then f i form a basis of V * . In particular, the restriction of Ψ to V 0 is a bijection with V * , hence an isomorphism. Now, applying the reasoning from the proof of Lemma 2.8 we find that each plot p of V 0 is of form p(u) = p 1 (u)v 1 + . . . + p k (u)v k for some smooth real-valued maps p 1 , . . . , p k . Hence the composition Ψ • p writes as (Ψ • p)(u) = p 1 (u)f 1 + . . . + p k (u)f k , which is obviously a plot of V * (the maps p 1 , . . . , p k being smooth, this is a plot for any vector space diffeology on V * ), therefore Ψ is smooth. Vice versa, arguing as in the proof of Theorem 2.12, every plot q of V * writes as q(u) = q 1 (u)f 1 + . . . + q k (u)f k for some smooth real-valued maps q 1 , . . . , q k . Thus, the composition Ψ −1 •q is of form (Ψ −1 •q)(u) = q 1 (u)v 1 +. . .+q k (u)v k , which is a plot of V 0 . We conclude that Ψ is smooth with smooth inverse, hence the conclusion.
The induced metric on V * It is now easy to see that any pseudo-metric on V induces a true metric on V * , via the smooth surjection Ψ described in Theorem 2.13. Let us describe this precisely; notice first of all that if f ∈ V * then there is a single element v 0 ∈ Ψ −1 (f ) such that v 0 ∈ V 0 . Furthermore, every other element in Ψ −1 (f ) is of form v 0 + v 1 , where v 1 ∈ V ⊥ , the orthogonal of the whole V with respect to the pseudo-metric ·|· A (equivalently, the subspace V 1 spanned by all the eigenvectors relative to the eigenvalue 0). Let now g ∈ V * be another element of the dual V * , and let w 0 + w 1 ∈ Ψ −1 (g).
The first equality implies that the pushforward of the pseudo-metric ·|· A to V * is well-defined (it could have been replaced by a reference to [7], Proposition 3.12 (1), of which it is a partial instance); the second equality means that this pushforward is a true metric. We summarize this discussion in the following statement.
Corollary 2.14. Any pseudo-metric ·|· A on V induces a true metric on the diffeological dual V * of V , via the natural pairing that assigns to each v ∈ V the smooth linear functional ·|v A .
Using the reasoning employed so far, we can also establish the inverse of this statement, namely, that every smooth metric (scalar product) on V * is induced by a pseudo-metric on V . Indeed, let ·|· B be smooth metric on V * , given by a k×k matrix B. Let {f 1 , . . . , f k } be an orthonormal basis of V * composed of eigenvectors of B. 9 As has already been noted, the form k i=1 f i ⊗ f i gives a certain pseudo-metric ·|· A on V ; it remains to check that this pseudo-metric does induce ·|· B on V * , and this follows from the fact that {f 1 , . . . , f k } is an orthonormal basis.

A simple example of a pseudo-metric
Let us return to an above example, considering V = R 3 endowed with the vector space diffeology generated by the plot p : R → V given by p(x) = |x|(e 2 + e 3 ). Observe first that the diffeological dual of V is generated by maps e 1 and e 2 − e 3 (with {e 1 , e 2 , e 3 } being the canonical basis of the usual dual of R 3 ). In particular, dim(V * ) = 2.
It is also easy to see that any smooth symmetric bilinear form on V is given by a matrix of form for some a, b, c ∈ R (the vector (0, |x|, |x|) must be orthogonal to any other vector of R 3 with respect to the associated bilinear form). For this matrix to define a pseudo-metric, we must have b, c > 0 and a 2 < bc.
To give a specific example, we can take a = 1, b = c = 2, obtaining A = We can however easily find that the subspace V 0 can be (better) described as V 0 = Span(e 1 , e 2 − e 3 ); the restriction of ·|· A to V 0 , with respect to the basis {e 1 , e 2 − e 3 } has the matrix (and yields indeed a true matrix). We also note that the subspace V 1 (generated by eigenvectors relative to 0) is Span( ).
Finally, let us calculate the matrix of the induced metric on V * with respect to its basis {e 1 , e 2 − e 3 }. For this, we need to find their pre-images with respect to Ψ| V0 (we'll just write Ψ for brevity). By easy calculation Ψ −1 (e 1 ) = 2 3 e 1 − 1 6 (e 2 − e 3 ) and Ψ −1 (e 2 − e 3 ) = − 1 3 e 1 + 1 3 (e 2 − e 3 ). Calculating their pairwise products with respect to the pseudo-metric ·|· A , we find the matrix 1 9 6 5 5 6 . 9 Note that the basis {f 1 , . . . , f k } generates the standard diffeology on V * , in the sense that each plot q : U → V * is locally of form q(u) = q 1 (u)f 1 + . . . + q k (u)f k for some smooth functions q 1 , . . . , q k : U → R. Indeed, each plot does write in this manner, simply by virtue of {f 1 , . . . , f k } being a basis; furthermore, ·|· B being smooth, and each constant map being a plot, implies that q(u)|f i B = q i (u) is smooth in the usual sense as a map U → R.