Fréchet Spaces and Presheave Morphisms.

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A topological vector space V is both a topological space and a vector space such that the vector space operations are continuous. A topological vector space is locally convex if its topology admits a basis consisting of convex sets (a set A is convex if (1 – t) + ty ∈ A ∀ x, y ∈ A and t ∈ [0, 1].

We say that a locally convex topological vector space is a Fréchet space if its topology is induced by a translation-invariant metric d and the space is complete with respect to d, that is, all the Cauchy sequences are convergent.

A seminorm on a vector space V is a real-valued function p such that ∀ x, y ∈ V and scalars a we have:

(1) p(x + y) ≤ p(x) + p(y),

(2) p(ax) = |a|p(x),

(3) p(x) ≥ 0.

The difference between the norm and the seminorm comes from the last property: we do not ask that if x ≠ 0, then p(x) > 0, as we would do for a norm.

If {pi}{i∈N} is a countable family of seminorms on a topological vector space V, separating points, i.e. if x ≠ 0, there is an i with pi(x) ≠ 0, then ∃ a translation-invariant metric d inducing the topology, defined in terms of the {pi}:

d(x, y) = ∑i=1 1/2i pi(x – y)/(1 + pi(x – y))

The following characterizes Fréchet spaces, giving an effective method to construct them using seminorms.

A topological vector space V is a Fréchet space iff it satisfies the following three properties:

  • it is complete as a topological vector space;
  • it is a Hausdorff space;
  • its topology is induced by a countable family of seminorms {pi}{i∈N}, i.e., U ⊂ V is open iff for every u ∈ U ∃ K ≥ 0 and ε > 0 such that {v|pk(u – v) < ε ∀ k ≤ K} ⊂ U.

We say that a sequence (xn) in V converges to x in the Fréchet space topology defined by a family of seminorms iff it converges to x with respect to each of the given seminorms. In other words, xn → x, iff pi(xn – x) → 0 for each i.

Two families of seminorms defined on the locally convex vector space V are said to be equivalent if they induce the same topology on V.

To construct a Fréchet space, one typically starts with a locally convex topological vector space V and defines a countable family of seminorms pk on V inducing its topology and such that:

  1. if x ∈ V and pk(x) = 0 ∀ k ≥ 0, then x = 0 (separation property);
  2. if (xn) is a sequence in V which is Cauchy with respect to each seminorm, then ∃ x ∈ V such that (xn) converges to x with respect to each seminorm (completeness property).

The topology induced by these seminorms turns V into a Fréchet space; property (1) ensures that it is Hausdorff, while the property (2) guarantees that it is complete. A translation-invariant complete metric inducing the topology on V can then be defined as above.

The most important example of Fréchet space, is the vector space C(U), the space of smooth functions on the open set U ⊆ Rn or more generally the vector space C(M), where M is a differentiable manifold.

For each open set U ⊆ Rn (or U ⊂ M), for each K ⊂ U compact and for each multi-index I , we define

||ƒ||K,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, ƒ ∈ C(U)

Each ||.||K,I defines a seminorm. The family of seminorms obtained by considering all of the multi-indices I and the (countable number of) compact subsets K covering U satisfies the properties (1) and (1) detailed above, hence makes C(U) into a Fréchet space. The sets of the form

|ƒ ∈ C(U)| ||ƒ – g||K,I < ε

with fixed g ∈ C(U), K ⊆ U compact, and multi-index I are open sets and together with their finite intersections form a basis for the topology.

All these constructions and results can be generalized to smooth manifolds. Let M be a smooth manifold and let U be an open subset of M. If K is a compact subset of U and D is a differential operator over U, then

pK,D(ƒ) := supx∈K|D(ƒ)|

is a seminorm. The family of all the seminorms  pK,D with K and D varying among all compact subsets and differential operators respectively is a separating family of seminorms endowing CM(U) with the structure of a complete locally convex vector space. Moreover there exists an equivalent countable family of seminorms, hence CM(U) is a Fréchet space. Let indeed {Vj} be a countable open cover of U by open coordinate subsets, and let, for each j, {Kj,i} be a countable family of compact subsets of Vj such that ∪i Kj,i = Vj. We have the countable family of seminorms

pK,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, K ∈  {Kj,i}

inducing the topology. CM(U) is also an algebra: the product of two smooth functions being a smooth function.

A Fréchet space V is said to be a Fréchet algebra if its topology can be defined by a countable family of submultiplicative seminorms, i.e., a countable family {qi)i∈N of seminorms satisfying

qi(ƒg) ≤qi (ƒ) qi(g) ∀ i ∈ N

Let F be a sheaf of real vector spaces over a manifold M. F is a Fréchet sheaf if:

(1)  for each open set U ⊆ M, F(U) is a Fréchet space;

(2)  for each open set U ⊆ M and for each open cover {Ui} of U, the topology of F(U) is the initial topology with respect to the restriction maps F(U) → F(Ui), that is, the coarsest topology making the restriction morphisms continuous.

As a consequence, we have the restriction map F(U) → F(V) (V ⊆ U) as continuous. A morphism of sheaves ψ: F → F’ is said to be continuous if the map F(U) → F'(U) is open for each open subset U ⊆ M.

Emancipating Microlinearity from within a Well-adapted Model of Synthetic Differential Geometry towards an Adequately Restricted Cartesian Closed Category of Frölicher Spaces. Thought of the Day 15.0

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Differential geometry of finite-dimensional smooth manifolds has been generalized by many authors to the infinite-dimensional case by replacing finite-dimensional vector spaces by Hilbert spaces, Banach spaces, Fréchet spaces or, more generally, convenient vector spaces as the local prototype. We know well that the category of smooth manifolds of any kind, whether finite-dimensional or infinite-dimensional, is not cartesian closed, while Frölicher spaces, introduced by Frölicher, do form a cartesian closed category. It seems that Frölicher and his followers do not know what a kind of Frölicher space, besides convenient vector spaces, should become the basic object of research for infinite-dimensional differential geometry. The category of Frölicher spaces and smooth mappings should be restricted adequately to a cartesian closed subcategory.

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Synthetic differential geometry is differential geometry with a cornucopia of nilpotent infinitesimals. Roughly speaking, a space of nilpotent infinitesimals of some kind, which exists only within an imaginary world, corresponds to a Weil algebra, which is an entity of the real world. The central object of study in synthetic differential geometry is microlinear spaces. Although the notion of a manifold (=a pasting of copies of a certain linear space) is defined on the local level, the notion of microlinearity is defined absolutely on the genuinely infinitesimal level. What we should do so as to get an adequately restricted cartesian closed category of Frölicher spaces is to emancipate microlinearity from within a well-adapted model of synthetic differential geometry.

Although nilpotent infinitesimals exist only within a well-adapted model of synthetic differential geometry, the notion of Weil functor was formulated for finite-dimensional manifolds and for infinite-dimensional manifolds. This is the first step towards microlinearity for Frölicher spaces. Therein all Frölicher spaces which believe in fantasy that all Weil functors are really exponentiations by some adequate infinitesimal objects in imagination form a cartesian closed category. This is the second step towards microlinearity for Frölicher spaces. Introducing the notion of “transversal limit diagram of Frölicher spaces” after the manner of that of “transversal pullback” is the third and final step towards microlinearity for Frölicher spaces. Just as microlinearity is closed under arbitrary limits within a well-adapted model of synthetic differential geometry, microlinearity for Frölicher spaces is closed under arbitrary transversal limits.