Functional analytic issues in $\mathbb{Z}_2^n$-Geometry

We show that the function sheaf of a $\mathbb{Z}_2^n$-manifold is a nuclear Fr\'echet sheaf of $\mathbb{Z}_2^n$-graded $\mathbb{Z}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a $\mathbb{Z}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of $\mathbb{Z}_2^n$-manifolds. All proofs are self-contained and explicit.


Introduction
Z n 2 -Geometry is an emerging framework in mathematics and mathematical physics, which has been introduced in the foundational papers [CGP16a] and [COP12]. This non-trivial extension of standard supergeometry allows for Z n 2 -gradings, where Z n 2 = Z ×n 2 = Z 2 × . . . × Z 2 and n ∈ N .
The corresponding Z n 2 -commutation rule for coordinates (u A ) A with degrees deg u A ∈ Z n 2 does not use the product of the (obviously defined) parities, but the scalar product −, − of Z n 2 : The definitions of Z n 2 -manifolds and Z n 2 -morphisms are recalled in Section 2. A survey of Z n 2 -Geometry is available in [Pon16].
The motivations for this new setting originate in both, mathematics and physics.
In physics, Z n 2 -gradings and the Z n 2 -commutation rule (n ≥ 2) are used in various contexts. References include [AS17], [AKTT16], [Tol13], and [YJ01], as well as [CKRS17] and [Kho16] (the latter, which can be traced back to [BB04], [DVH02], and [MH03], use the Z n 2 -formalism implicitly). Further, the Z n 2 -commutation rule is not only necessary, but also sufficient: it can be shown [CGP14] that any commutation rule, for any finite number m of coordinates, is of the form (1), for some n ≥ 2m.
In mathematics, well-known algebras are Z n 2 -commutative. This holds for instance for the algebra of quaternions (which is Z n 2 -commutative with n = 3) and, more generally, for any Clifford algebra Cl p,q (R) (Z n 2 -commutative with n = p + q + 1), as well as for the algebra of Deligne differential forms on a standard supermanifold (Z n 2 -commutative with n = 2: the first Z 2 -degree is induced by the cohomological degree, the second Z 2 -degree is the parity).
Moreover, there exist canonical examples of Z n 2 -manifolds. The local model of these higher or colored supermanifolds is necessarily [Pon16] the Z n 2 -commutative algebra C ∞ (x) [[ξ]] of formal power series in the coordinates ξ = (ξ 1 , . . . , ξ N ) of non-zero Z n 2 -degree, with coefficients in the smooth functions in the coordinates x = (x 1 , . . . , x p ) of zero Z n 2 -degree. For instance, the tangent bundle of a classical Z 2 -manifold or supermanifold M can be viewed as a Z 2 2manifold T M, in which case the function sheaf is the completion of Deligne differential forms of M. Actually, the tangent (and cotangent) bundle of any Z n 2 -manifold is a Z n+1 2 -manifold. Moreover, the 'superization' of any double vector bundle (resp., any n-fold vector bundle) is canonically a Z 2 2 -manifold (resp., Z n 2 -manifold). We expect that a number of applications of Z n 2 -Geometry in physics are based on the integration theory of Z n 2 -manifolds. A first step towards Z n 2 -integration is the Z n 2 -generalization of the Berezinian. This fundamental concept has been constructed in [COP12] and is referred to as the Z n 2 -Berezinian. The Z n 2 -integration theory is still under investigation. Other applications of Z n 2 -Geometry rely on Z n 2 Lie groups (generalized super Lie groups) and their actions on Z n 2 -manifolds (which are expected to be of importance in supergravity), on has a right-adjoint Hom A (V, −) : Z n 2 Mod(A) → Z n 2 Mod(A) , i.e., for any U, W ∈ Z n 2 Mod(A), there is a natural isomorphism where Hom A (V, W ) denotes the categorical hom made of the degree-respecting A-linear maps, i.e., of the A-linear maps ℓ : V → W of degree 0 ∈ Z n 2 , i.e., ℓ(V γ ) ⊂ W γ+0 , for all γ ∈ Z n 2 . One can readily verify that the internal hom Hom A (V, W ) is the Z n 2 -graded A-module which consists of all A-linear maps ℓ : V → W of all possible degrees α ∈ Z n 2 , i.e., for all γ ∈ Z n 2 . The A-linear maps of degree α constitute the α-part Hom α A (V, W ) of Hom A (V, W ). Hence, contrary to the case of modules over a classical commutative algebra, the internal hom Hom A differs from the categorical hom Hom A , since this latter contains only 0-degree A-linear maps: Hom A (V, W ) = Hom 0 A (V, W ).

Sheaves of differential operators on a Z n 2 -manifold
It is known that bump functions and partitions of unity do exist in Z n 2 -manifolds [CGP14], and that they can be used similarly to classical bump functions in smooth geometry [Lei80].
Notice that A = R can be viewed as a Z n 2commutative algebra concentrated in degree 0, and that, for any open U ⊂ M , the algebra O(U ) is an object in Z n 2 Mod(R), i.e., is a Z n 2 -graded R-vector-space. Hence, according to Section 3, is the Z n 2 -graded R-vector-space of all R-linear maps of all Z n 2 -degrees from O(U ) to itself. Composition endows this space with a Z n 2 -graded associative unital R-algebra structure, and the Z n 2 -graded commutator [−, −] endows this space with a Z n 2 -graded Lie algebra structure. We can identify O(U ) with an associative subalgebra of End R (O(U )) using the left-regular representation g → m g , m g (f ) = g · f .
Definition 4. The Z n 2 -graded O(U )-module of k-th order differential operators D k (U ), k ∈ N, is defined inductively by where This implies by induction that any differential operator of any order is local.
be a bump function of M around m with support supp γ ⊂ V and restriction γ| W = 1, for some neighborhood W ⊂ V of m. It then follows from the defining property of differential operators applied to [D, γ]f , the induction assumption, and the fact γf = 0, that (Df )| W = 0.
We now show that any D ∈ D k (U ) can be localized. Proof. If f ∈ O(V ) and m ∈ V , we choose a function F ∈ O(U ) such that F | W = f | W , for some neighborhood W ⊂ V of m (it suffices to choose a bump function γ in M around m with support in V and restriction 1 to W (m ∈ W ⊂ V ) and to set F = γf ). Locality of D implies that the section (DF )| W ∈ O(W ) and the section (DF ′ )| W ′ ∈ O(W ′ ) obtained similarly but for a point m ′ ∈ V , depend only on f and thus coincide on the intersection W ∩ W ′ . Hence, these local sections define a unique global section as announced. Since, obviously, D| V ∈ End R (O(V )) (note that D| V has the same parity as D), it suffices -to prove Proposition 5 -to observe that, for any with obvious notation.
where r U V is the restriction map of the function sheaf O), and they satisfy the usual compatibility conditions for restriction maps. The presheaf D k is in fact a sheaf over M -the standard proof goes through. If we wish to emphasize the base topological space of this sheaf we write  [CGP14]. Moreover, Z n 2 -morphisms are J -adically continuous and the induced morphisms between stalks are m-adically continuous [CGP14]. Moreover: Proposition 6. Any differential operator ∆ ∈ D k (U ) of any degree k ≥ 0 and over any open U ⊂ M is J (U )-adically continuous. In particular, if a sequence of functions f n ∈ O(U ) tends J (U )-adically to a function f ∈ O(U ), then ∆f n tends J (U )-adically to ∆f . Due to its locality, any differential operator ∆ of order k acting on O(U ) canonically induces, for any m ∈ M , a differential operator ∆ m of order k acting on O m , and this induced operator ∆ m is m m -adically continuous.
The proof is based on the following lemmata. We will omit the subscript M which remembers that most of the sheaves considered here are defined over M .
for all c ≥ 1.
Proof. In this proof we just write ∆ thus omitting the superscript indicating its order. If k = 0, the differential operator ∆ is a function ∆ ∈ O(U ). Hence, for any c ≥ 1, Assume now that the claim holds for k ≥ 0. We will show that it is then also valid for k + 1. Let c = 1, let ∆ ∈ D k+1 (U ), let f ∈ J (U ) and let g ∈ J k+1 (U ). We have f ∆g ∈ J (U ), since J (U ) is an ideal, and, since [∆, f ] ∈ D k (U ), we have [∆, f ]g = ∆(f g) ± f ∆g ∈ J (U ), in view of the induction assumption. It follows that ∆(f g) ∈ J (U ) and that ∆ J k+2 (U ) ⊂ J (U ): for differential operators of order k + 1 the claim is true for the lowest value c = 1. It thus suffices to assume that the statement holds for c ≥ 1 and to prove that it holds then as well for c + 1. Therefore, consider f ∈ J (U ) and g ∈ J k+c+1 (U ). Since [∆, f ]g = ∆(f g) ± f ∆g ∈ J c+1 (U ), due to the induction assumption on k, and since f ∆g ∈ J c+1 (U ), due to the induction assumption on c, we finally obtain that ∆ J k+c+2 (U ) ⊂ J c+1 (U ), what completes the proof.
The next lemma is similar and has the same proof. We are now prepared to prove Proposition 6.
) is actually open. The claim that the limit of ∆f n is the value of ∆ at the limit of f n , is a direct consequence of Lemma 7. The statement concerning ∆ m follows analogously from Lemma 8.
Remark 10. Note that the decomposition of a differential operator of order k in this local basis leads to a finite sum ( see Equation (7) below ).
Proof. Since we work locally, we take M = (U, , where U is an open subset of R p . We first prove uniqueness of the coefficients. If D ∈ D k (U ) is of the type and if m αβγ = 1 α!β! x α θ β ζ γ , where the coordinates are written in increasing order and |α| + |β| + |γ| = i, then necessarily Indeed, the term with same indices as the considered monomial reduces to its coefficient when applied to the monomial, and any other term contains at least one index that is higher than the corresponding index in the monomial, so that this term annihilates the monomial). Hence, the coefficients D i αβγ of D are unique, if they exist. More precisely, for |µ| + |ν| + |π| = 1 and |α| + |β| + |γ| = 2, we get D 0 000 = Dm 000 , Consider now an arbitrary D ∈ D k (U ) and set ∆ = D − Σ ∈ D k (U ), where Σ denotes the RHS of (7) with the coefficients defined in (8). This operator ∆ vanishes by construction on the polynomials of degree ≤ k in x, θ, ζ. The reader might wish to check this claim by direct computation. For instance, the computation for k = 2 is straightforward in view of Equation (9).
Note now that, for any f 1 , . . . , as immediately seen when developing (10) shows that ∆ = 0 on any polynomial of degree k + 1 in x, θ, ζ. Indeed, for coordinate functions f 1 , . . . , f k , h, the value ∆(f 1 . . . f k h) vanishes if and only if F (1) vanishes. However, the value F (1) is, again in view of (10), computed from values of ∆ on polynomials of degree ≤ k and does therefore vanish. It now follows by induction from (10) that ∆ = 0 on an arbitrary polynomial in x, θ, ζ. We can extend this conclusion to polynomial formal series, i.e., to series where µ is a multi-index with sum of components denoted by |µ|, and where P µ (x) is a polynomial in the zero-degree coordinates x. Indeed, the sequence P c obtained by taking in the series P only the terms |µ| < c converges J (U )-adically to the series P, since in local coordinates the ideal J c (U ) is made of the series all whose terms contain at least c formal coordinates. Hence, Proposition 6 implies that ∆P ∈ J c (U ), for all c, i.e., that and let m be any point in U . By the polynomial approximation theorem [CGP14, Theorem 6.10], there exists, for any c, a polynomial formal series P such that [f ] m − [P] m ∈ m k+c m (P depends on f, m, and c). When applying the differential operator where Σ is, as above, the RHS of Equation (7).
The definition of D 1 (U ) implies straightforwardly that 2 -graded associative and Z n 2 -graded Lie algebra structures that have weight 0 and −1, respectively, with respect to the filtration degree. The assignment D : U → D(U ) is a locally free sheaf of Z n 2 -graded O-modules and of Z n 2 -graded associative and Lie algebras over M . The algebra D(M ) is the Z n 2 -graded Lie algebra of differential operators of the Z n 2 -manifold M.
Remark 11. The reader observed probably that (as usual) D k−1 (U ) ⊂ D k (U ). Hence, a k-th order differential operator is actually a differential operator of order ≤ k. To emphasize this fact some authors write D ≤k (U ) instead of D k (U ).

Functional analytic properties of the function sheaf of a Z n 2manifold
For a review of Fréchet spaces, algebras, and sheaves, we refer the reader to the Appendix.
• A Z n 2 -graded Fréchet vector space is a Z n 2 -graded vector space V , all whose homogeneous subspaces V γ , γ ∈ Z n 2 , are Fréchet vector spaces. We denote by (p γ i ) i∈I a family of seminorms corresponding to V γ .
• A Z n 2 -graded nuclear LCTVS is a Z n 2 -graded vector space V , all whose homogeneous subspaces V γ , γ ∈ Z n 2 , are nuclear.
such that there are equivalent countable families of seminorms that are submultiplicative, i.e., equivalent countable families (q γ n ) n∈N of seminorms, such that, for all n ∈ N, algebras over a smooth manifold M , such that all section spaces F(U ) are Z n 2 -graded Z n 2 -commutative (nuclear) Fréchet algebras and the locally convex topology on F(U ) is the coarsest topology for which all restriction maps F(U ) → F(U i ) are continuous.
The algebraic direct sum ⊕ α V α of a family (V α ) α∈A of LCTVS-s V α is usually equipped with the direct sum topology, that is, with the finest locally convex topology such that the injections i α : V α → ⊕ α V α are all continuous. In this case, we refer to the direct sum as the topological direct sum of the LCTVS-s V α . It is known that a countable topological direct sum of nuclear LCTVS-s is a nuclear LCTVS. Further, a finite topological direct sum of Fréchet spaces is a Fréchet space. These results show that a Z n 2 -graded Fréchet vector space (resp., a Z n 2 -graded nuclear LCTVS) is a Fréchet space (resp., a nuclear LCTVS), when equipped with the direct sum topology.
Remark 13. In the following, we suppress the superscript γ in the various seminorms p γ i , q γ n , ... that we consider. In other words, p i , q n , ... refer to a seminorm of some space V γ .
We are now prepared to prove one of the main theorems of this paper.
Proof. Let U ∈ Open(M ), let C be any compact subset of U , and let D be any differential operator in D(U ). For any f ∈ O(U ), we set where ε is the projection ε : Proof. It suffices to prove the separability. If sup x∈C |ε(D(f ))(x)| = 0, for all C and all D, then ε(D(f )) = 0 in U for any D, since U admits a (countable) cover by compact subsets C, see Lemma 32. Differently stated, we have for any D ∈ D k (U ) and for any k ∈ N. Let now (V i ) i∈N be a (countable) cover of U by Z n 2 -chart domains (any open cover of U admits a countable subcover, see proof of Lemma 32) and let V be any element of this cover. For any m ∈ V , there is a Z n 2 -bump-function γ ∈ O 0 (U ) and neighborhoods N 1 ⊂ N 2 ⊂ V of m such that γ| N 1 = 1 and supp γ ⊂ N 2 . In view of Equation for any D ∈ D k (U ) and any k ∈ N.
The next lemma covers the case where U is a Z n 2 -chart domain. Lemma 16. Let U ⊂ M be a Z n 2 -chart domain with coordinates u = (x, ξ).
be a sequence of functions (resp., a function) in O(U ). The sequence f n is Cauchy in O(U ) ( resp., converges to f in O(U ) ) if and only if the sequences f nβ are all Cauchy in C ∞ (U ) ( resp., converge all to the corresponding f β in C ∞ (U ) ).
• The locally convex Hausdorff space O(U ) is complete.
• The space O(U ) is a Z n 2 -graded nuclear Fréchet algebra.
We start with the following observation. Let (U, u = (x, ξ)) be a Z n 2 -coordinate system, D ∈ D k (U ), and f ∈ O(U ). We have If β = γ, either there is β i > γ i , or all β i ≤ γ i but for at least one i we have β i < γ i . In the first case ∂ β ξ ξ γ = 0 and in the second ∂ β ξ ξ γ ∈ J (U ), so that in both situations the corresponding term in the series over γ vanishes under the action of ε. If β = γ, the derivative with respect to ξ equals β! , so that We are now prepared for the proof of Lemma 16.
Proof. • Assume first that the f nβ are all Cauchy in the locally convex topology of C ∞ (U ): for any base differential operator ∆ (acting on base functions C ∞ (U )) and any compact C ⊂ U , we have if r, s → ∞, see Example 34. In this case, we get, for any Z n 2 -differential operator D (acting on Z n 2 -functions O(U )) and any compact C ⊂ U , if r, s → ∞, so that f n is Cauchy in the topology of O(U ). Conversely, if f n is Cauchy in O(U ), we have to show that (15) holds for any ∆, C, and β. Fix these three data. The base differential operator ∆ reads ∆ = α ∆ α (x) ∂ α x and D = ∂ β ξ ∆ is a Z n 2 -differential operator. In view of (14), we get The proof of the convergence statement of item one in Lemma 16 is similar.
• In view of Example 34 and item one, item two is obvious.
• To prove that the complete Hausdorff locally convex topological vector space O(U ) is a Fréchet space, it suffices to show that there exists a countable family of seminorms on O(U ) that is equivalent to the family (p C,D ) C,D . To prove that C ∞ (U ) is a Fréchet space, one uses a countable cover of U by compact subsets C n ⊂ U such that C n is contained in the interior of C n+1 [Tre67]. Proceeding similarly, we define the family where the components of α belong to N and the components of β to N or Z 2 depending on whether they correspond to even degree or odd degree parameters. Since the p Cn,α,β are specific p C,D , they are a countable family of seminorms on O(U ). To show that the countable family is equivalent to the original one, we use Proposition 31. For any p C,D there exists C n ⊃ C, so that The similar condition for p C,D and p Cn,α,β exchanged is obviously satisfied, since any p Cn,α,β is a specific seminorm of the type p C,D .
Finally the Z n 2 -graded vector space O(U ) is a Fréchet space. This should of course mean that O(U ) is a Z n 2 -graded Fréchet vector space in the sense of Definition 12. In the paragraph following that definition, we mentioned that, if the homogeneous subspaces O γ (U ), γ ∈ Z n 2 , are Fréchet, then O(U ) is Fréchet as well. A rigorous application of Definition 12 requires now that we take an interest in the converse result. However, what we proved so far is valid for any functions in O(U ), in particular for the functions of a fixed degree, i.e., for the functions in O γ (U ), γ ∈ Z n 2 . It follows that all spaces O γ (U ), γ ∈ Z n 2 , are Fréchet spaces for the seminorms considered. Hence, the space O(U ) is a Z n 2 -graded Fréchet space in the sense of Definition 12. Alternatively, the reader may observe that any subspace of a Fréchet space, which contains the limits of its converging sequences, is itself a Fréchet space. Indeed, the restrictions to this subspace of the countable and separating family of seminorms of the total space is again a countable and separating family of seminorms. In view of Proposition 27, the resulting locally convex seminorm topology of the subspace is implemented by a translation-invariant metric. To be Fréchet, the subspace must still be complete with respect to this metric, i.e., it has to be complete with respect to the seminorm topology. Now, any Cauchy sequence of the subspace is Cauchy in the total space and converges therefore in the total space. But, by assumption, its limit is located in the subspace, so that the subspace is complete with respect to its topology. In the case considered here, any homogeneous subspace O γ (U ) of the Fréchet space O(U ) contains the limits of its converging sequences (in view of point 1 of the preceding lemma) and is thus Fréchet (so we can conclude again that O(U ) is a Z n 2 -graded Fréchet vector space in the sense of Definition 12).
It remains to show that O(U ) is a Z n 2 -graded nuclear LCTVS in the sense of Definition 12. Since any subspace of a nuclear space is nuclear, it suffices to prove that the locally convex space O(U ) is nuclear. We set q = (q ′ , q ′′ ), where q ′ = (q 1 , . . . , q 2 n−1 −1 ) and q ′′ = (q 2 n−1 , . . . , q 2 n −1 ) give the number of parameters in each nonzero even degree and the number of parameters in each odd degree, respectively. We also set A = N |q ′ | × Z |q ′′ | 2 . Further, we consider the coordinate order (6) and we order the monomials ξ α = θ β ζ γ using the lexicographic order with respect to α = (β, γ). This leads to a linear vector space isomorphism We identify O(U ) with α∈A C ∞ (U ) via i, so that i = id. Since C ∞ (U ) is nuclear, see Example 6.2.2, and since any product of nuclear LCTVS-s is a nuclear LCTVS for the product topology, the space O(U ) is nuclear for this topology. It is known that, if π b : a∈A V a → V b is a product of LCTVS-s V a , whose locally convex topologies are implemented by families of seminorms (ρ a i ) i , then the locally convex product topology is implemented by the family of seminorms (ρ a i ) a,i defined byρ a i = ρ a i • π a . Hence, in the case considered here, the product topology is given by the family of seminormsp α ∆,C = p ∆,C •π α . Of course, the standard locally convex topology on O(U ), i.e., the seminorm topology of the family p C,D , must coincide with the product topology, i.e., the families of seminormsp α ∆,C and p C,D must be equivalent. Let therefore β ∈ A, f ∈ O(U ), let ∆ be a differential operator acting on C ∞ (U ), and let C be a compact subset of U . When noticing that D = 1 β! ∂ β ξ ∆ is a differential operator acting on O(U ), we obtaiñ Conversely, for any differential operator D acting on O(U ), we have Corollary 17. For any open subset Ω ⊂ R p , the map where the source ( resp., the target ) is equipped with the standard topology induced by (p C,D ) C,D ( resp., the product topology of the standard topologies induced by (p ∆,C ) ∆,C ), is an isomorphism of TVS-s.
The next lemma will allow us to almost complete the proof of Theorem 14.
if n → ∞, for all compact subsets C ⊂ U and all Z n 2 -differential operators D on U , if and only if, for all i, if n → ∞, for all compact subsets C i ⊂ U i and all Z n 2 -differential operators D i on U i . Assume first that (21) holds, let C i , D i be as said, and consider a bump function γ ∈ O(U ) that equals 1 in an open neighborhood V of C i and is compactly supported in U i . Since C i is a compact subset of U and since γD i is a Z n 2 -differential operator on U , we get if n → ∞. Conversely, if (22) holds and if C, D are as above, there exists a finite open cover (V j ) j of C such that eachV j is compact and eachV j is contained in some U i(j) [CCF11]. Then, It remains to show that the Z n 2 -graded associative R-algebra O(U ) is a Z n 2 -graded Fréchet algebra for its standard Hausdorff locally convex topology given by the family 2 -graded Fréchet algebra, we have to provide an equivalent countable submultiplicative family of seminorms on O(U ). It actually suffices to proceed as above, see (18). More precisely, for each one of the countably many Z n 2 -chart domains U i , we can choose a countable cover of U i by compact subsets C n,i ⊂ U i such that C n,i is contained in the interior of C n+1,i . The family . We will omit the subscripts j, i in the arbitrarily chosen U j,i ⊂ φ −1 (V j ), as well as subscript j in V j , and we will write ν n for the restriction f n | V . We denote by u = (u A ) = (x a , ξ a ) the coordinates in U and by v = (v B ) = (y b , η b ) the coordinates in V . Let now C ⊂ U be a compact subset and let D = α D α (u)∂ α u be a differential operator acting on O(U ). We have to prove that The Z n 2 chain rule [CKP16] where the sum and products are finite, where C denotes real numbers, and where we limited ourselves to the structure of this complex result. It follows that the supremum in (23) reads where we omitted the restrictions to U and where F ∈ O(U ). We get Since ν n → 0 in O(V ) and φ(C) is a compact subset of V , the conclusion follows.
In order to extend the conclusion from φ * N to φ * V , for any open V ⊂ N , note that we can restrict the Z n 2 -morphism . It suffices to set ϕ = φ| U : U → V , and to set, for any open W ⊂ V , Indeed, the base map ϕ is smooth and the pullbacks ϕ * W are Z n 2 -graded unital R-algebra morphisms, which commute with restrictions. Applying now the first part of our proof of Item 2, we get that ϕ *

Appendix
In this section, we recall basic results and provide examples.

Definitions and construction
Remark 20. All vector spaces considered in the present text are spaces over the field R of real numbers.
Definition 21. A topological vector space (TVS) is locally convex if its topology has a basis made of convex subsets, i.e., subsets U such that, for any x, y ∈ U , the segment {(1 − t)x + ty : Definition 22. A locally convex topological vector space (LCTVS) is a Fréchet space if its topology can be implemented by a translation-invariant metric with respect to which it is (sequentially) complete.
Recall that a metrizable TVS is complete if and only if it is sequentially complete.
The standard construction of a Fréchet space starts from a family of seminorms. The difference between a seminorm p on a vector space and a norm || − || is that p(x) = 0 does not imply that x = 0. Recall also that a family of seminorms (p i ) i∈I separates points (or is separating), if for x = 0, there is i ∈ I such that p i (x) = 0.
The following proposition is almost obvious and will not be proven.
Proposition 23. Let (p i ) i∈I be a family of seminorms on a vector space V . For any i ∈ I, x ∈ V, ε > 0, set The family of all the subsets B i (x, ε) generates a topology on V ( recall that this topology is made of the unions of finite intersections of subsets B i (x, ε) ). We refer to this topology as the seminorm topology induced by (p i ) i∈I . The finite intersections of subsets B i (x, ε) with fixed x and ε, form a basis B of the seminorm topology: The elements of this basis are convex. The open subsets U of the seminorm topology are characterized by the property that for any x ∈ U , there is a basis element ∩ n j=0 B i j (x, ε), which is contained in U . The seminorm topology endows V with a structure of LCTVS.
Proposition 24. Let (p i ) i∈I be a family of seminorms on a vector space V . A sequence (x k ) k∈N of elements of V converges to x ∈ V with respect to the seminorm topology, if and only if it converges to x with respect to each seminorm, i.e., if and only if Similarly, the sequence (x k ) k∈N is a Cauchy sequence with respect to the seminorm topology, if and only if it is Cauchy with respect to each seminorm.
Proof. We prove the second statement. Let (x k ) k∈N be Cauchy with respect to the topology and take i ∈ I and ε > 0.
so that the neighborhoods are disjoint.
As well-known, the next definition of Fréchet spaces is equivalent to the above one, but is better suited for applications.
Definition 26. A TVS is a Fréchet space if it is Hausdorff and (sequentially) complete, and if its topology can be induced by a countable family of seminorms.
We are now prepared to discuss standard construction methods of Fréchet spaces from countable families of seminorms.
Proposition 27. If (p n ) n∈N is a countable family of seminorms on a vector space V , and if this family separates points, then is a translation-invariant metric on V that induces the seminorm topology of V .
Proof. The statement is a standard functional analytical result.
The next proposition is natural. It extends Proposition 24: Proposition 28. In the situation of Proposition 27, a sequence in V converges to a limit in V (resp., is a Cauchy sequence) with respect to the metric d, if and only if it converges (resp., is a Cauchy sequence) with respect to the seminorm topology, and if and only if it converges (resp., is a Cauchy sequence) with respect to all seminorms.
Remark 29. To construct a Fréchet space, one usually proceeds as follows, although the method admits a number of variants. On starts with a vector space V and a countable and separating family (p n ) n∈N of seminorms on it. The seminorm topology turns V into a Hausdorff LCTVS. Definition 26 then allows us to conclude that V is a Fréchet space, if we can prove that V is (sequentially) complete with respect to its topology. In view of Proposition 28, this condition is equivalent to (sequential) completeness with respect to the translation-invariant metric (25) induced by the seminorms. Further, if one can verify that a sequence in V that is Cauchy for any seminorm p n , converges to a fixed x ∈ V for any p n , then the TVS V is (sequentially) complete with respect to the seminorm topology, again due to Proposition 28.

Examples
We briefly present some examples.
Example 30. The vector space R ∞ of all sequences r = (r 0 , r 1 , . . .) of real numbers is a Fréchet space for the countable family (p n ) n∈N of seminorms p n (r) = sup m≤n |r m | .
The family (q n ) n∈N given by q n (r) = r≤n |r m | defines the same topology, i.e., the same Fréchet space. We say that two such families of seminorms are equivalent.
We recall an important criterion for equivalence of two families of seminorms.
Proposition 31. To families of seminorms (p i ) i∈I and (q j ) j∈J on a vector space V are equivalent, i.e., they induce the same locally convex topology, if and only if, for any i, there is a constant C > 0 and a finite subset {j 1 , . . . , j N } ⊂ J, such that and vice versa. Any open subset Ω ⊂ R p admits a cover ∪ x∈Ω B(x) by open balls B(x) whose adherencē B(x) is contained in Ω. In view of the Lindelöf property, we can extract from the preceding open cover of Ω a countable subcover The latter cover is searched countable cover by compact subsets.
Let now U ⊂ M be an open subset of a p-dimensional smooth manifold. We can cover U by coordinate systems (U α , ϕ α ) and extract a countable subcover (U i , ϕ i ). Since ϕ i is a homeomorphism between U i ⊂ M and ϕ i (U i ) ⊂ R p , the set U i admits a countable cover U i = ∪ k∈N C ki by compact subsets C ki ⊂ U i . We thus get the countable cover U = ∪ i∈N ∪ k∈N C ki of U by compact subsets C ki ⊂ U .
The following is one of the important examples of Fréchet spaces.
Example 33. For any open subset Ω ⊂ R p , the function algebra C ∞ (Ω) is a Fréchet vector space for the countable family (p α,i ) α,i of seminorms defined, for any multi-index α ∈ N ×p and any compact C i (i ∈ N) of a countable cover of Ω by compact subsets (e.g., C i may run through the ballsB(x i ) of (26)), by p α,i (f ) = sup To prove this standard statement one uses Remark 29. The result can be extended: Example 34. The function algebra C ∞ (U ) of an open subset U of a (second-countable Hausdorff finite-dimensional) smooth manifold M , is a Fréchet vector space. The locally convex topology of C ∞ (U ) is implemented (for instance) by the family of seminorms where ∆ ∈ D(U ) is any differential operator acting on C ∞ (U ) and where C is any compact subset of U . Note that -by definition -this topology is the topology of uniform convergence on compact subsets C of f and its 'derivatives' ∆(f ).

Definition
Let us recall that a completion of a TVS V is a complete TVSV that contains V (or, better, a homeomorphic image of V ) as a dense subspace. Any (LC)TVS can be completed as When considering (algebraic) tensor products of LCTVS-s, some subtleties arise due to the possibility to choose various topologies on these products.
More precisely, let V, W be two LCTVS-s. The finest locally convex topology on the algebraic tensor product V ⊗ W , for which the natural map V × W → V ⊗ W is continuous, is referred to as the projective tensor topology. The completion of the resulting LCTVS is the completed projective tensor product V ⊗ π W . There exists another natural locally convex topology on V ⊗ W , which is coarser than the projective one, and which is called the injective tensor topology. The corresponding completion is the completed injective tensor product V ⊗ i W . Any reasonable locally convex topology on V ⊗ W lies between the injective and projective ones.
We are now prepared to give one of the equivalent definitions of nuclear LCTVS-s.
More precisely, the identity id : V ⊗ π W → V ⊗ i W is a bijective continuous linear map, and its continuous extension id : V ⊗ π W → V ⊗ i W is an injective continuous linear map. When V is nuclear, this canonical map is onto, or, better, it is a TVS-isomorphism. As already said, any (reasonable) locally convex topology on V ⊗ W is located between the projective and the injective tensor topologies. Hence, if V is nuclear, the complete TVS V ⊗W is independent of the locally convex topology considered.
Nuclear Fréchet spaces are just a specific type of nuclear LCTVS-s. Fréchet spaces are a full subcategory of TVS-s, and so are nuclear spaces.

Example
When thinking about the duality between spaces and function algebras, one meets the problem of interpreting a tensor product of function algebras as function algebra of some space. Even in the case of algebras C ∞ (Ω) of smooth functions on open subsets Ω of Euclidean spaces, the canonical map C ∞ (Ω ′ ) ⊗ C ∞ (Ω ′′ ) → C ∞ (Ω ′ × Ω ′′ ) is (of course) not an isomorphism. However, if one endows the algebraic tensor product of the LCTVS-s C ∞ (Ω ′ ) and C ∞ (Ω ′′ ) with the projective tensor topology and considers the corresponding completion, one gets an isomorphism of TVS-s: The topology that we choose on the algebraic tensor product is actually irrelevant -a space of the type C ∞ (Ω) is nuclear. More precisely, both, C ∞ (Ω) (Ω ⊂ R p ) and C ∞ (U ) (U ⊂ M , M smooth manifold), are nuclear Fréchet spaces.
In more detail, if V, W are complete LCTVS-s and if V is a concrete space (e.g., V = C ∞ (Ω ′ )), then it is mostly impossible to characterize both V ⊗ π W and V ⊗ i W concretely (in fact a space of bilinear forms on dual spaces of V and W is also involved here, but we refrain from describing this space precisely). For example, when V = C ∞ (Ω ′ ) and W = C ∞ (Ω ′′ ), we can interpret C ∞ (Ω ′ ) ⊗ i C ∞ (Ω ′′ ) concretely as the space C ∞ (Ω ′ × Ω ′′ ), but we have no good access to C ∞ (Ω ′ ) ⊗ π C ∞ (Ω ′′ ). If we know a priori that V = C ∞ (Ω ′ ) is nuclear, the problem disappears.

Fréchet algebras
In fact, the algebra C ∞ (U ), where U is any open subset of any smooth manifold, is a Fréchet algebra [MH05]. The definition of a Fréchet algebra is natural: Definition 36. A Fréchet algebra is a Fréchet vector space A, which is equipped with an associative bilinear and (jointly) continuous multiplication · : A × A → A. If (p i ) i∈I is a family of seminorms that induces the topology on A, (joint) continuity is equivalent to the existence, for any i ∈ I, of j ∈ I, k ∈ I, and C > 0, such that p i (x · y) ≤ C p j (x) p k (y), ∀x, y ∈ A.
We can always consider an equivalent increasing countable family of seminorms (|| − || n ) n∈N . The preceding condition then requires that, for any n ∈ N, there is r ∈ N, r ≥ n and C > 0, such that ||x · y|| n ≤ C ||x|| r ||y|| r , ∀x, y ∈ A.
In particular, the topology can be induced by a countable family of submultiplicative seminorms, i.e., by a family (p n ) n∈N that satisfies p n (x · y) ≤ p n (x) p n (y), ∀n ∈ N, ∀x, y ∈ A.
Note that many authors define a Fréchet algebra simply as a Fréchet vector space, which carries an associative bilinear multiplication, and whose topology can be induced by a countable family (q n ) n∈N of submultiplicative seminorms. This latter definition is equivalent to the former.

Fréchet sheaves
Let M be a smooth manifold and denote by Open(M ) (resp., Alg(R)) the category of open subsets of M (resp., of associative R-algebras). As mentioned above, the function sheaf is actually valued in nuclear Fréchet algebras, i.e., in nuclear Fréchet vector spaces that carry a Fréchet algebra structure. In view of this observation, it is natural to consider Fréchet sheaves. Their definition is well-known: Definition 37. A sheaf F of (real) vector spaces over a smooth manifold M is a Fréchet sheaf of vector spaces, if the next two conditions are satisfied: • for any U ∈ Open(M ), the vector space F(U ) is a Fréchet vector space, and • for any U ∈ Open(M ) and any cover (U i ) i∈I of U , U i ∈ Open(U ), the locally convex topology on F(U ) is the coarsest topology for which the restriction maps F(U ) → F(U i ) are continuous.
Since F is a sheaf of (real) vector spaces, it follows from the second condition that, for any V ∈ Open(U ), the restriction map F(U ) → F(V ) is R-linear and continuous. As Fréchet spaces are a full subcategory of TVS-s, the second requirement of Definition 37 is thus quite natural. In view of this understanding, it is clear that the definition of a (nuclear) Fréchet sheaf of algebras is similar, but starts from a sheaf of (real) algebras.