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Владикавказский математический журнал 2012, Том 14, Выпуск 4, С. 73-82
517.983
USING HOMOLOGICAL METHODS ON THE BASE OF ITERATED SPECTRA IN FUNCTIONAL ANALYSIS
E. I. Smirnov
We introduce new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Frechet spaces and is stable under forming countable inductive and projective limits, closed subspaces and quotient spaces. Moreover, for H-space an unproved variant of the closed graph theorem holds true. Homological methods are used for proving of theorems of vanishing at zero for first derivative of Hausdorff limit functor: Haus1 (X) = 0.
Key words: topology, spectrum, closed graph theorem, differential equation, homological methods, category.
Introduction
The study which was carried out in [1-2] of the derivatives of the projective limit functor acting from the category of countable inverse spectra with values in the category of locally convex spaces made it possible to resolve universally homomorphism questions about a given mapping in terms of the exactness of a certain complex in the abelian category of vector spaces. Later in [3] a broad generalization of the concepts of direct and inverse spectra of objects of an additive semiabelian category G (in the sense V. P. Palamodov) was introduced: the concept of a Hausdorff spectrum, analogous to the ¿^-operation in descriptive set theory. This idea is characteristic even for algebraic topology, general algebra, category theory and the theory of generalized functions. The construction of Hausdorff spectra X = {Xs, F, hss} is achieved by successive standard extension of a small category of indices The category H of Hausdorff spectra turns out to be additive and semiabelian under a suitable definition of spectral mapping. In particular, H contains V. P. Palamodov's category of countable inverse spectra with values in the category TLG of locally convex spaces [1]. The H-limit of a Hausdorff spectrum in the category TLG generalizes the concepts of projective and inductive limits and is defined by the action of the functor Haus : H ^ TLC. The class of H-spaces is defined by the action of the functor Haus on the countable Hausdorff spectra over the category of Banach spaces; the closed graph theorem holds for its objects [8] and it contains the category of Frechet spaces and the categories of spaces due to De Wilde [7], D. A. Rajkov [5] and Suslin [6]. The H-limit of a Hausdorff spectrum of H-spaces is an H-space [7]. There are many injective objects in the category H and the right derivatives Haus* (i = 1,2,...) are defined, while the "algebraic" functor Haus : H (L) ^ L over the abelian category L of vector spaces (over R or C) has injective type, that is if 0 ^ X ^ Y ^ Z is an exact sequence of mappings of Hausdorff spectra with values in L, then the limit sequence 0 ^ Haus(X) ^
© 2012 Smirnov E. I.
Haus(Y) ^ Haus(Z) is exact or acyclic in the terminology of V. P. Palamodov [2]. In particular, regularity of the Hausdorff spectrum X of the nonseparated parts of Y guarantees the exactness of the functor Haus : H(TLC) ^ TLC and the condition of vanishing at zero: Haus*(X) = 0. The classical results of Malgrange and Ehrenpreis on the solvability of the unhomogeneous equation p(D)D' = D', where p(D) is a linear differential operator with constant coefficients in Rn and D' = D'(S) is the space of generalized functions on a convex domain S C Rn, can be extended to the case of sets S which are not necessarily open or closed. Analogous theorems for Frechet spaces were first proved by V. P. Palamodov [1-2].
1. We recall certain definitions and theorems which are used in this chapter and which were brought into the discussion in [3-6]. Let 0 be a small category. By a directed class in the category we mean a subcategory satisfying the following properties:
(i) no more than one morphism is defined between any two objects;
(ii) for any objects a, b there exists an object c such that there exist a ^ c and b ^ c.
Let A be some category and s denotes the object of a category A (if Q G 0 and a, b G Q
we will denote the corresponding morphisms of category 0 by a —^ b). We shall call the category B with objects S, where S is a subcategory of A, a standard extension of the category A if the following conditions are satisfied:
1°. A is a complete subcategory of B;
2°. The morphism uss' : S' ^ S of the category B is defined by the collection of morphisms uss' : s' ^ s (s' s) of the category A such that
(a) for every s' G S' there exists s G S such that s' s;
(b) if s' —^ s, p' p, s —^ p, then there exists a morphism s' —^ p' and the following diagram is commutative:
S
s -> p
/ S' / s' -> p/
We will establish the successive standard extensions of categories
fi(s) C B(T) C E(F) ^ S0(F) C D(F),
where T C ^ denotes directed classes of objects s G fi, coincides as object of category B; F, F G B, denote filter bases of sets T G B, considered as objects of category S, and F, F C S, denote directed classes of objects F G S of the dual category S0, considered as objects of category D. We shall say that such classes F are admissible for fi; put |F| = (JTgF T, |F| = (JfgF |Fso that |F| C fi and |F| C fi. The most characteristic constructions connected with Hausdorff spectra use in the role of the small category fi = Ord I, where I is a partially ordered set of indices, considered as category.
Example 1 (standard extension of the category A). Let G and A be categories, T(F) the category of covariant functors F : G ^ A with functorial morphism $ : Fi ^ F2 defined by the rule [2] which assigns to each object g G G a morphism $(g) : F1(g) ^ F2 (g) of the category A such that for any morphism u : g ^ h of the category G the following diagram is commutative
Fi(h) -—U F2 (h)
Fi(w)
F2H
Fi (g) -Î^-U F2(g)
w
It is clear that each object s £ A generates a covariant functor Fs : g £ G ^ s £ A such that A c T. Moreover, A is a complete subcategory of T.
We will show that T provides a standard extension of the category A (by means of the category G). Let F £ T and S c A be such that S = UgeGF(g) and for s', s £ S the set of morphisms Hom(s',s) = UwF(w), where w : g ^ h and s' = F(q), s = F(h). Therefore the category B is defined, where S is a subcategory of A and the morphisms wSS' : S' ^ S of the category B are generated by the collection of functorial morphisms $ : F' ^ F, where F' £ T generates S', while F generates S according to the method indicated above.
If we take such a functorial morphism $ : F' ^ F, then the morphisms $(g) : F'(g) ^ F(g) (g £ G) of the category A form a collection of morphisms wss' : s' ^ s (s' = F' (g), s = F(g)) such that (a) is satisfied. Condition (b) follows from consideration of the definition of the functorial morphism.
Thus, B is a standard extension of the category A. If G = Ord I, where I is a linearly ordered set, then T = B(S).
Example 2 (Palamodov [1]). The categories of direct and inverse spectra over a semia-belian category K are standard extensions of the category K.
Example 3 (construction of an admissible class for fi). Let T be a separated topological space and fi a countable set. We shall call a set A c T an s-set if
A = U n T.,
BeK ieB
where T. (t £ fi) is a subset of T and K is the family of subsets B of the set fi such that
(a) for each B £ K the set TB = n.eBT. is compact in T,
(b) the sets TB (B £ K) form a fundamental system of compact subsets of A.
Proposition 1. Every separable metric space is an s-set.
Proposition 2. Let A be a subset of the finite-dimensional space Rn. Then A is an s-set and moreover
A = U n^ (1)
BeK ieB
where the T. are compact subsets of Rn.
Thus, s-sets are a generalization on the one hand of compact spaces (and locally compact spaces which are countable at infinity) and on the other of separable metric spaces. However, s-sets will be of interest to us in connection with the possibility of constructing the associated functor of a simple Hausdorff spectrum.
Let A be some s-set, so that
A = T.,
BeK ieB
where T. c T, B c fi. We may assume without loss of generality that the family Q of subsets T. (t £ fi) is closed with respect to finite intersections and unions (that is, there exist corresponding surjections $s, : d(fi) ^ fi, where d(fi) is the set of finite subsets of fi).
The set fi will be partially ordered if we put t' ^ t whenever T. c T.'; let G = OrdQ. Further, we may assume that each set B £ K is directed in (fi, Let I be the factor set of all possible complexes s = [ti, t2,..., tn], where t* £ |K |, t* = pr*s (i = 1,2, ...,n, n £ N), with respect to the equivalence relation on the set of ordered n-tuples of elements of |K | : (ti,t2 ,...,t„) ~ (t/1,t/2,...,tn) if and only if {ti,t2 ,...,tn} = {ti, t'2,..., 4}. The set I becomes partially ordered if we put s' ^ s, where s = [t1, t2,..., tn], s' = [t/, t2,..., t^], whenever for each t* there exists tj such that tj ^ t*; let fi = Ord I.
By continuing the construction following the method of transformation of indices we will construct an admissible class F for fi. For each s = [ti, t2,...,tn] £ |F| the subset Rs = (Jn=i Tti is defined and moreover if s' ^ s then Rs c Rs'. Thus a contravariant functor of the simple Hausdorff spectrum H(A) : |F | ^ G is defined and moreover
A = U HRs-
(2)
FSFsSF
It is an essential point that I is a countable set and the family { P|f Rs} is a fundamental system of nonempty compact subsets of A.
Let G be some category. We shall call a covariant functor Hf : fi ^ G a Hausdorff spectrum functor if fi = |F | for some admissible class F £ D. If F = |F | then Hf is a functor of the direct spectrum, while if F = {|F|} (that is, F consists of a single element |F| = |F|) then HF is a functor of the inverse spectrum.
If F is an admissible class for fi and the functor
Hf : <
|F| ^ G, s ^ Xs,
(s' —— s) ^ (Xs ^ Xs>),
(F' F) ^ ((Xs)sS|F| ^ (Xs')s'S|F'|)
is injective on objects and morphisms (in the set-theoretic sense), then there exists a directed class ( )
((Xs)se|F|,qFF')F,F' £ F
of classes (Xs, Hs's)s s'S|F| (F £ |F|) which are directed in the dual category G0 and which satisfy the following conditions.
1°. The morphism Xs Xs' is chosen and fixed if and only if the morphism s' —^ s is chosen and then Hs's : Xs ^ Xs' is the only morphism. 2°. The diagram
Xs
Xs
Xs
Xs
f 11 '' ^s's'' ' wss'
is commutative for all s —> s —> s.
3°. If (Xs)sS|F| ——"F (Xs')s's|f'|, then for each Xs' (s' £ |F'|) there exists a unique morphism Hs's : Xs ^ Xs (s £ |F|). The collection of morphisms Hs's (s' £ |F'|) defines the morphism qF'F so that we shall write qF'F = (Hs's)F'F. Each set F £ F is a filter base of subsets T c |F| and moreover for each T £ F the class (Xs,Hs's)T is directed in the category G0.
Definition 1. We shall call a class (Xs, Hs's)s s'S|F| satisfying conditions 1°-3° a Hausdorff spectrum over the category G and we shall denote it by {Xs, F, Hs's}.
The direct and inverse spectra of a family of objects are particular cases of Hausdorff spectra: it suffices to put F = |F|, Hs's = qF'F in the direct case and F = {|F|}, Hs's : Xs ^ Xs' (s' ^ s), qF'F = i|F| = i|F| in the inverse case.
Under a suitable definition of spectral mapping (see the structure of the category D(F)) the set of Hausdorff spectra over G forms a category which we denote by Spect G. If X =
h
h
hs's''
's
{Xs, F, hs's}, Y = {Yp, F are objects from Spect G, then we shall say that two
Hausdorff spectrum mappings : X u Y and WyX : X u Y are equivalent if for any F £ F there exists F* £ F1 such that the diagram
Xs
hp*p
Yp' -► Yp*
is commutative for any p* £ |F*
Now let us consider a new category H(G) whose objects are the objects of the category Spect G, but the set HomH (X, Y) is formed by the equivalence classes of mappings : X u Y. We shall denote such classes by ||wyx||.
For any objects X, Y, Z £ H the law of composition defines a bilinear mapping
HomH (X, Y) x HomH (Y, Z) u HomH (X, Z)
(HomH(X, Y) is an abelian group).
Definition 2. Let X = {Xs, F, hs's} be a Hausdorff spectrum over the category G. We shall call an object Z of the category G a categorical H-limit of the Hausdorff spectrum X over G if for any objects A, B £ G and spectral mappings A —U X —> B there exists a
unique sequence in G A
Z
ß
B such that the diagram A ^^ X
Z
ß
B
is commutative in the category Spect G.
The concepts of projective and inductive limits over the category G are special cases of categorical H-limits. For example, let X be the inverse spectrum of objects from G. Then (Lim) holds and moreover any object Xs from X can be taken for B £ G with the identity morphism bs : Xs u Xs forming the spectral mapping bs : X u Xs (s £ |F|). Thus the following diagram is commutative
A ^^ X
ß
X
Z
where b = (bs), ft = (^s), : Z u Xs (s £ |F|), b is the identity morphism of the category Spect G. Therefore the diagram
A ^^ X
Z
ß
X
is commutative for any object A £ G.
The categorical H-limit of a Hausdorff spectrum (the functor Haus) exists in any semi-abelian category G with direct sums and products (for example, the category of vector
p s
b
a
b
a
a
spaces L, the category TLC of topological vector groups, the category TLC of locally convex spaces).
Let 0 be a countable set and X = {Xs,F, hs's} a regular Hausdorff spectrum in the category TLC; such a spectrum is said to be countable. A continuous linear image in the
category TLC of an H-limit X = lim^ hs'sXs of Banach spaces Xs (s £ |F |) is called an
F
H-space. The class of H-spaces contains the Frechet spaces and is stable with respect to the operations of passage to countable inductive and projective limits, closed subspaces and factor spaces. Moreover, a strengthened variant of the closed graph theorem holds for H-spaces. The class of H-spaces is the broadest of all the analogous classes known at this time, namely those of Rajkov, De Wilde, Hakamura, Zabrejko-Smirnov. A countable separated regular H-limit of a Hausdorff spectrum of H-spaces in the category TLC is an H-space [7].
Throughout this chapter Hausdorff spectra are assumed to be countable unless the contrary is explicitly stated.
2. Let Haus : H(TLC) 4 L be the covariant additive Hausdorff limit functor from the semiabelian category H(TLC) to the abelian category L of vector spaces (over R or C). We recall [11] that by an injective resolvent I of an object X £ H(TLC) we mean any sequence
0 Io Ii ...,
formed by injective objects and exact in its members I k, k ^ 1, with ker i0 = X. Any two injective resolvents of the same object are homotopic to each other. Since there are many injective objects in the category H(TLC) [3], each object of this category has at least one injective resolvent. The right derivatives of the Hausdorff limit functor Haus are defined by the formula
Hausk(X) = Hk(Haus(I)) (k = 0,1,...),
where X £ H(TLC), I is any injective resolvent of X, Haus(I) is the complex of morphisms of the category L obtained by application of the functor Haus to each morphism of the complex I, and Hk(Haus(I)) (k = 0,1,...) are the homologies of the complex Haus(I). Each morphism X 4 Y of the category H(TLC) is covered by a morphism I 4 Y of the injective resolvents of the objects X and Y (see [11, Chapter V, § 1]). From this follows the existence of morphisms Hausk (X) 4 Hausk (Y) so that the objects of Hausk (X) do not depend on the choice of injective resolvent. On the other hand the functor Haus has injective type [3, p. 88], therefore the canonical isomorphism of functors holds:
Haus = Haus0.
Proposition 3. For every free Hausdorff spectrum E £ H(L)
Haus^(E) = 0 (i = 1,2,...).
We now compute the derived functors Haus* (i ^ 1) in the following way (see [2, 10]). Let X = {Xs, F, hs's} be an arbitrary Hausdorff spectrum and E the free Hausdorff spectrum with generators Xs (s £ |F|). Let us consider the sequence of Hausdorff spectrum mappings
0 X —4 E E 0, (D)
in which the components of the mapping (i.e. the collection (wTsT)T)|, where
st £ T is the unique maximal element in T with respect to the direction relation) act
according to the formula wTst : xsT u (hs'sTxsT)s'eT, while the Hausdorff spectrum mapping wee : E u E is formed by means of the morphisms (Tn is a cofinal right-filtering sequence)
WT*Tn : (xs)sSTn U (xs* - hs*sT„XsTJs*eT*
for any T*, T„ £ F, F £ F, To = 0, Tra_i C T* C Tra, STn C T* (n = 1, 2,...).
It is now clear that the sequence (D) is exact; following V. P. Palamodov [2] we shall call the sequence (D) the canonical resolvent of the Hausdorff spectrum X.
Applying the functor Haus to the canonical resolvent (D) we obtain the sequence of locally convex spaces
0 u Haus(X) J]Xs J]Xs,
F F F F
where 0fT!f Xs is the direct sum of the products of the Xs (s £ |F |) under the natural inductive limit topology; this sequence is acyclic and moreover exact from the left. Proposition 4. Let Haus : H(TLC) u L and let
0->X -— Y Z-.0 (D')
be an exact sequence of Hausdorff spectra. Then the following exact connecting sequence is defined in the category L (^1 (i = 1,2,...) are the connecting morphisms):
0 —► Haus(X) —► Haus( Y) —► Haus(Z) —► Haus1 (X)
-u ... -u Haus (Z) —u Haus*(X) —U Haus^( Y) --U Haus*(Z) ...
3. In [1] and [2] V. P. Palamodov established the fundamental Theorems 11.1 and 11.2 giving necessary and sufficient conditions for the vanishing at zero Pro1 (X) = 0 for the functor Pro of the projective limit of a countable family of locally convex spaces. We aim to establish analogous conditions for the vanishing at zero Haus1 (X) = 0 for the Hausdorff limit functor and for the not necessarily countable case.
We recall that in questions concerning the stability of the class of H-spaces with respect to Hausdorff limits and also in the theorem about the representation of H-spaces by means of Banach spaces the assumption of regularity of the Hausdorff spectrum was an important condition. Here it will be necessary to impose the following condition. Let X = {Xs, F, hs's}T be a Hausdorff spectrum of locally convex spaces and for each T £ F let C nF Xs be defined by
vj = jx = (xs) £ J^Xs : Xs' = hs'sXs, s,s' £ t|,
equipped with the projective topology with respect to the preimages n_1Ts (s £ T), where ns : nf Xs u Xs is the canonical projection. The corresponding; base of neighborhoods of zero for the projective topology generates the TVG (Xs,tf(T)) (T £ F).
Let us form the TVG (HF Xs,a(F)) with base of neighborhoods of zero VF (T £ F). The Hausdorff spectrum X is said to be regular if (nFXs,a(F)) satisfies the condition: convergence of a net (a7)7eP in the TVGs(nF Xs,a(T)) (T £ F) implies its convergence in the TVG (nF Xs,a(F^. If every Xs (s £ |F|) has the indiscrete topology, then it is not difficult to see that the first part of the condition for regularity is equivalent to completeness of (nF Xs,^(F)).
Theorem 1. Let X be a regular Hausdorff spectrum of nonseparated parts over the category TLC. Then Haus1 (X) = 0.
If Y is a regular Hausdorff spectrum over TLC and X is the Hausdorff spectrum of nonseparated parts, then it is easy to see that X is also a regular spectrum. In fact, bearing in mind the remark before the theorem, it is sufficient to establish the completeness of ( ]1f C(.f)); this TVG is embedded in the corresponding TVG (nF Ys, )) • If (aY)ySp is fundamental under CT(F), then aY G aY0 + V^ (V T G F, 7 >- y(T), 70 >- 7(T)) and because of the closedness of V^ in the latter TVG we obtain the inclusion (a* = limP a7)
a* - a7o G Vj (VT G F, 70 ^ 7(T)),
which also implies the convergence of (aY) to a* in (Xs,a(F))•
Thus, in the enunciation of Theorem 1 regularity of the Hausdorff spectrum X can be replaced by regularity of the Hausdorff spectrum Y.
Theorem 2. Let Y be a regular Hausdorff spectrum, X the Hausdorff spectrum of non-separated parts of Y and 0 ^ X ^ Y ^ Y/X ^ 0 an exact sequence of Hausdorff spectra. Then the sequence 0 ^ Haus(X) ^ Haus( Y) ^ Haus( Y/X) ^ 0 is exact in the category L.
Let us continue our consideration of the question of exactness of the functor Haus : H(TLC) ^ L for an arbitrary exact sequence of Hausdorff spectra 0 ^ X ^ Y ^ Z ^ 0. From the proofs given above it is clear that a sufficient condition for the vanishing at zero Haus1 (X) = 0 is the completeness of the TVG(HF Xs , ct(f)) for each F G F (see Proposition 7.1 of [3]), where I(F) is formed by the filtering V^ with respect to T. At the same time each space V^ is endowed with the linear topology defined by the inverse image supTn-Vs (T G F) forming at the same time the TVG (nF Xs, CT(F)) so that the TVG inFXs,CT(F)) is not in general metrizable. It turns out that completeness of the TVG( nF , ^(F)) is also a necessary condition for the vanishing at zero Haus1 (X) = 0.
Proposition 5. Let X = {Xs, F, hs's} be a countable Hausdorff spectrum over the category L. Then in order that Haus1 (X) = 0 it is necessary and sufficient that the TVG(nF a(F)) is complete for each F G F.
Theorem 3. Let X = {Xs, F, hs's} be a countable Hausdorff spectrum over the category L. Then in order that Haus1(X) = 0 it is necessary and sufficient that for each F G F it is possible to define in F Xs a quasinarm ^ = ^ 0 such that
(i) the associated topologiaal group (Xs,r(F)) is complete, tf ^ a(F),
(ii) ^F is continuous on (Xs,ct*f)).
< Necessity. This follows from the argument before the theorem, since on putting tf = c*F) and
^f (x) = dTk (x),
fc=1
where dTk (x) = 0 for x G V^ and dTk (x) = 1 for x G nF Xs ^F* (k G N), we obtain (i) and (ii).
Sufficiency. Let ZF = P|^=1 VFk and let the factor space F Xs/ZF be endowed with the images of the topologies a(F) and tf, so that, if
^
dF (£) = inf (x) and dF (£) = inf dTk (x),
k=1
the MVG(HF Xs/ZF, dF) is separated and complete and the MVG(Hf Xs/ZF,dp) is separated. Thus on the MVG(HF Xs/ZF, dF) the functional dF is countably semiadditive and
dF (0 = inf lim dF (£n) = inf pF (x)
is continuous on it. Hence by the Lemma on a countably semiadditive functional [8] we obtain dF = dF and, consequently, the MVG(HF Xs/ZF,dF) is complete. But this means that the F)) will be complete, which allows us to conclude on considering all F G F that Haus^X) = 0. The Theorem is proved. >
In the case of a countable inverse spectrum, in particular, we obtain the first part of Theorem 11.1.1 of [1]; in the case of a direct spectrum X the topology тF is indiscrete for each singleton set F G F. Moreover, the famous lemma of V. P. Palamodov [1], which makes up the main part of the proof, is a special case of the lemma about a countably semiadditive functional [8].
In what follows ^F denotes the filter topology on Xs (s G |F|), which is formed by the spaces {hss'Xsi} (s' G |F|). We note, however, that the product topology on ПF Xs obtained from the topologies ^>F (s G |F|) does not in general coincide with the topology ).
Sufficient conditions for the vanishing at zero
Haus1 (X) = 0, which are more convenient for applications, are given in the following proposition.
Theorem 4. Let X = {Xs, F, hs's} be a countable Hausdorff spectrum over the category L. In order that Haus1(X) = 0 it is sufficient that for each s G |F| it is possible to define in Xs a family of quasinorms {pes} which determines a complete separated pseudotopological vector space (Xs,p^s), preserves the continuity of the morphisms hss and is such that for each s G |FF G F the following condition is satisfied:
(A) for some в = es(F) the functional p^ is continuous in the filter topology (Xs, ^>F)•
In particular, in the case of an inverse spectrum X we obtain Theorem 5.1 of [2] and moreover our assertion is even stronger in this case.
Theorem 5. Let X = {Xs, F, hs's} be a countable Hausdorff spectrum of separated H-spaces over the category TLC. Then in order that Haus1 (X) = 0 it is necessary and sufficient that the spaces (Xs, ^>F) (s G |F |) are complete TVGs for each F G F.
In the case of an inverse spectrum of Frechet spaces Theorem 5 extends the criteria (F) and (R) of V. P. Palamodov's Corollary 11.4 in [1]. We note that in Theorem 5 it is separatedness of the pseudotopology which is actually required, therefore in general the H-space may be nonseparated.
Theorem 6. Let X = {Xs, F, hs's} be a countable Hausdorff spectrum of H-spaces over the category TLC with separated associated pseudotopology {(pfs)*} which preserves the continuity of the morphisms hsrs. Then in order that Haus1 (X) = 0 it is necessary and sufficient that for each s G |F| there exists a quasinorm pPs (F) (s G |F|) in Xs such that
(A') (pps)* is continuous in the filter topology ^>F and the system {pps} preserves the continuity of the morphisms hs's.
In particular the theorem by Retakh [9] follows from Theorem 6.
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Received September 12, 2012. Smirnov Eugeny Ivanovich
Pedagogical University, Department of Mathematics, Professor, Ph. D. Hab., Head of Calculus Department 108 Respublikanskaya Avenue, Yaroslavl, 150000, Russia; Southern Mathematical Institute Vladikavkaz Science Center of the RAS,
Leading Researcher of Laboratory of Educational Technologies 22 Markus street, Vladikavkaz, 362027, Russia E-mail: [email protected]
ИСПОЛЬЗОВАНИЕ ГОМОЛОГИЧЕСКИХ МЕТОДОВ НА БАЗЕ ИТЕРИРОВАННЫХ СПЕКТРОВ В ФУНКЦИОНАЛЬНОМ АНАЛИЗЕ
Смирнов Е. И.
В статье водятся новые понятия функционального анализа: хаусдорфов спектр и хаусдорфов предел или Н-предел хаусдорфова спектра в категории локально выпуклых пространств (или даже, в более общих полуабелевых категориях). Частными случаями регулярного хаусдорфова предела являются проективный и индуктивный пределы отделимых локально выпуклых пространств. Новый класс Н-пространств содержит пространства Фреше и замкнут относительно операций взятия счетного индуктивного и проективного пределов, перехода к замкнутому подпространству и фактор-пространству. Более того, для Н-пространств справедлив усиленный вариант теоремы о замкнутом графике. Доказаны теоремы об обращении в нуль первой производной функтора хаусдорфова предела средствами гомологической алгебры.
Ключевые слова: топология, спектр, замкнутый график, дифференциальные уравнения, гомологические методы, категория.
