Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними
We consider some classes of Frechet spaces with integrable derivatives. An important compactness lemma for nonreflexive spaces are obtained. Some main topological properties for the given spaces are obtained.
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| author | Kasyanov, P. Mel'nik, V. Piccirillo, A.-M. |
| author_facet | Kasyanov, P. Mel'nik, V. Piccirillo, A.-M. |
| author_institution_txt_mv | [
{
"author": "P. Kasyanov",
"institution": null
},
{
"author": "V. Mel'nik",
"institution": null
},
{
"author": "A.-M. Piccirillo",
"institution": null
}
] |
| author_sort | Kasyanov, P. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2018-04-11T11:11:24Z |
| description | We consider some classes of Frechet spaces with integrable derivatives. An important compactness lemma for nonreflexive spaces are obtained. Some main topological properties for the given spaces are obtained. |
| first_indexed | 2025-07-17T10:23:33Z |
| format | Article |
| fulltext |
© P. Kasyanov, V. Mel'nik , A.-M. Piccirillo, 2007
Системні дослідження та інформаційні технології, 2007, № 4 93
TIДC
НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ,
ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
UDC 517.19
ON SOME APPROXIMATIONS AND MAIN TOPOLOGICAL
DESCRIPTIONS FOR SPECIAL CLASSES OF FRECHET SPACES
WITH INTEGRABLE DERIVATIVES
P. KASYANOV, V. MEL'NIK , A.-M. PICCIRILLO
We consider some classes of Frechet spaces with integrable derivatives. Important
compactness lemmas for nonreflexive spaces are obtained. Some main topological
properties for the given spaces are obtained.
Method of monotony and method of compactness represent fundamental ap-
proaches to study nonlinear differential-operator equations, evolutionary inclu-
sions and variational inequalities in Banach spaces. The general idea is the fol-
lowing: using the corresponding approximation scheme, the approximate
solutions of a problem are constructed, for them the approaching a priori esti-
mates are established, at last they prove the existence of sequence of approximate
solutions, that converges to the exact solution of problem. In many cases the
aim is obtained by using both a method of compactness and a method of
monotonicity.
Now we introduce some constructions to prove the convergence of Faedo–
Galerkin method for a global solvability of differential-operational equations, in-
clusions and evolution variation inequalities with λw –pseudomonotone maps [1,
2, 3, 4]. Moreover, we obtain a new theorems of compact embedding for Frechet
spaces, suggested by researches of differential-operational inclusions in function
spaces.
For a pair of Banach spaces YX , the notation YX ⊂ further will mean the
embedding both in the set-theory sense and in the topological sense.
Let Y be some Banach space; *Y be its topologically conjugated space; I
be some compact time interval. We consider the classes of functions defined on I
and imagines in Y (or in *Y ).
The set );( YILp of measured by Bochner functions (see [5]) as +∞≤≤ p1
with the natural linear operations is a Banach space with the norm
.)(=);(
1/p
p
Y
I
pL dttyYIy
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∫
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 94
As +∞=p we have a Banach space );( YIL∞ with the norm
.)(maxess=);( Y
It
L tyYIy
∈∞
The next theorem proves that under the reflexivity or separability of Y the
conjugated to );( YILp , +∞≤ <1 p , *));(( YILp may be identify with
);( *YILq , 1=11 −− + qp .
Theorem 1. (Rietz) If the space Y is reflexive or separable and +∞≤ <1 p ,
then each element *));(( YILf p∈ gives the unique representation
);(everyfor)(),(=)( YILydttytyf pY
I
∈〉〈∫ ξ
with the function );( *YILq∈ξ , 1=11 −− + qp . The correspondence ξ→f ,
*));(( YILf p∈ is linear and
.);(= **));(( YIf
qLYIpL
ξ
Now let us consider the reflexive separable Banach space V with the norm
V⋅ and the Hilbert space )),(,( HH ⋅⋅ with the norm H⋅ , and let the next con-
ditions are valid:
.:0>
,indenseis,
Vvvv
HVHV
VH ∈∀≤∃
⊂
γγ
(1)
Under these assumptions we may consider the conjugated to H , *H , as a
subspace *V that is conjugated to V . As V is reflexive then *H is dense in *V
and
,*
** Hfff
HV
∈∀≤ γ
where *V
⋅ is the norm in *V , *H
⋅ is the norm in *H .
Further, we identify the spaces H and *H . Then we obtain ∗⊂⊂ VHV
with continuous and dense embedding.
Definition 1. The triple of spaces ( *;; VHV ), that satisfies the latter condi-
tions is called the evolution triple.
Let us note that under identification of H with *H and *H with some sub-
space of *V , an element Hy∈ coincides with some *Vf y ∈ and =),( xy
Vy xf 〉〈= , Vx∈∀ , where V⋅〉〈⋅, is the canonical pairing between *V and V .
Since the element y and yf are identified then, under condition (1), the pairing
V⋅〉〈⋅, and the inner product on H will be denoted the same notation ),( ⋅⋅ .
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 95
By the analogy with [7] we consider ip , ir , 1,2=i such that ≤ip<1
+∞≤≤ ir , +∞<ip . Let 1≥≥ ii rq well-defined defined by =+ −− 11
ii qp
1=11 −
′
− += ii rr 1,2=i∀ . Remark that 0=1/∞ .
Now we consider some Banach spaces that play an important role in the in-
vestigation on differential-operator equations and evolution variation inequalities
in non-reflexive Banach spaces.
For evolution triples ( *;; ii VHV ) ( 1,2=i ) such that
HVVV and,spacestheindenseissetthe 2121 ∩ (2)
and for some compact time interval we consider the functional Banach spaces
1,2=),;();(=)( * iHILVILIX
iriiqi ′
+
with norms
∈
′
12*1 |});(;);({max{inf=)( yHIyVIyIy
ir
LiiqLiX
}=),;(),;( 212
* yyyHILyVIL
iriiq +∈∈
′
,
for all )(IXy i∈ , and * *
1 21 2 2 1
( ) = ( ; ) ( ; ) ( ; ) ( ; )q q r rX I L I V L I V L I H L I H
′ ′
+ + +
with
);(|});(;);({max{inf= *
12*1
1,2=
)( iiqi
ir
Li
iiqLi
i
IX VILyHIyVIyy ∈
′
,
}=1,2;=),;( 222112112 yyyyyiHILy
iri +++∈
′
,
for each Xy∈ . We remark that if +∞<ir then the space )(IX i is reflexive.
Analogously, if +∞<},{max 21 rr , then the space )(IX is reflexive.
Following by [7] we identify )(* IX i , conjugated to )(IX i , with
( ; ) ( ; )r p ii i
L I H L I V∩ , where );();(=)(* iipL
ir
LiX VIyHIyIy + ∈∀ y
)(* IXi∈ and )(* IX , conjugated to )(IX , with ∩∩ );();(
21
HILHIL rr
);();( 2211
VILVIL pp ∩∩ , where ++ );();(=)( 21
* HIyHIyIy
rLrLX
)();();(
*
2211
IXyVIyVIy
pLpL ∈∀++ .
On )()( * IXIX × we denote the duality form by the rule:
++〉〈 ∫∫ ττττττ dyfdyfyf H
I
H
I
I ))(),(())(),((=, 1211
=)(),()(),(
222121 ττττττ dyfdyf V
I
V
I
〉〈+〉〈+ ∫∫
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 96
*,))(),((= XXfdyf
I
∈∈∀∫ τττ ,
where 22211211= fffff +++ , );(1 HILf
iri ′
∈ , );( *
2 iiqi VILf ∈ , 1,2=i .
Let 21= VVV ∩ , )(VF be a filter of all finite-dimensional subspaces from
V . As V is separable, there exists a countable monotone increasing system of
subspaces )(}{ 1 VH ii F⊂≥ complete in V , and consequently in H . On nH we
consider inner product induced from H , that we denote again as ),( ⋅⋅ . Moreover
let HHHP nn ⊂→: be the operator of orthogonal projection from H on nH :
.minarg=everyfor Hn
nnh
n hh
H
hPHh −
∈
∈
Definition 2. We say that the triple ( )HVH ii ;;}{ 1≥ satisfies condition (γ ),
if +∞
≥
<sup ),(
1
VVn
n
P L , i.e. there exists such 1≥C that for every Vv∈ and 1n ≥
.VVn vCvP ≤ (3)
Some constructions that satisfy the above condition were presented in [6].
Remark 1. It is easy to check that there exists such complete orthonormal in
H vector system Vh ii ⊂≥1}{ such that for every 1≥n nH is a linear capsule
stretched on n
iih 1=}{ . Then condition )(γ means that the system is a Schauder ba-
sis in the space V (see [9], p.403).
Remark 2. From the identification between *H and H it follows that *
nH
and nH are also identified.
Remark 3. In virtue of ),( VVPn L∈ for every 1≥n the conjugate operator
),( *** VVPn L∈ and )*,*(
*
),( = VVnVVn PP LL . It is obvious that for every
Hh∈ hPhP nn
*= . Hence, we identify nP with its conjugate *
nP for every 1≥n .
Then, the condition )(γ will mean that for every Vv∈ and 1≥n
.and ** VVnVVn vCvPvCvP ≤⋅≤ (4)
Let us denote by S a subset of a real line which can be presented as no more
than numerable join of convex sets in R . We denote by Θ∈αα }{=)( ISBC the
family of all convex bounded sets from S , distinct from a point.
Remark 4. Notice that )(= SΘΘ , i.e. the set of indexes depends on the set
S . Further, Θ will mean )(SΘ .
Furthermore we set
{ )}(||:= *loc
αα
α IXyVSyX I ∈Θ∈∀→ ,
where *
2
*
1
* = VVV + . In this space the local base of topology is the following:
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 97
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
≥Θ∈ 1,1,=,0,>),(=:
1=
nnkV kkkk
n
k
X αεεα∩B ,
where for every Θ∈α and 0>ε :
⎭
⎬
⎫
⎩
⎨
⎧ ∈ εεα
α
<)(|=),( *
*
loc IuXuV
X
.
Lemma 1 XB is local base of some topology Xτ in locX , which converts
the given space in a separable locally convex linear topological space and, more-
over,
a) Xτ is compatible with the set of seminorms
loc
( on})=)({ XIX Θ∈⋅⋅ αααρ ; (5)
b) a set locXE ⊂ is bounded only when Θ∈∀α αρ is bounded on E .
Proof. We prove the system of seminorms Θ∈ααρ }{ divides points on
locX . Let }0{\locXu∈ , then 0>)0)(|( ≠∈ tuStλ , where λ is Lebesgue meas-
ure on R . Because of S is a subset of a real line which can be presented as no
more than numerable join of convex sets in R , we have :0 Θ∈∃α
0>)
0
( αIXu . From here it follows 0>)(
0
uαρ , as it was to be shown.
From [12, theorem 1.37] it follows, that the system of seminorms Θ∈ααρ }{
generates some locally convex topology Xτ on locX , which converts the given
space in locally convex linear topological space, whose local base we obtain by
final intersections of such sets:
{ }0>,|}<)(|{=),( loc εαερε αα Θ∈∈ uXuIV .
The statement b) follows from the same theorem.
The lemma is proved.
Let
{ })(||:= ***
loc αα
α IXyVSyX I ∈Θ∈∀→ .
In this space the local base of topology is the following:
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
≥Θ∈ 1,1,=,0,>),(=:
1=
* nnkV kkkk
n
k
X
αεεα∩B
where for every Θ∈α and 0>ε :
⎭
⎬
⎫
⎩
⎨
⎧ ∈ εεα
α
<)(|=),( *
*
loc IuXuV
X
.
Lemma 2. *X
B is local base of some topology *X
τ in *
locX , which con-
verts the given space in a separable locally convex linear topological space and,
moreover,
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 98
a) *X
τ is compatible with the set of seminorms
*
loc* on})(=)({ XIX Θ∈⋅⋅ α
α
αρ ; (6)
b) a set *
locXE ⊂ is bounded only when Θ∈∀α αρ is bounded on E .
Proof. As well as in lemma 1 it is enough to show that system of seminorms
Θ∈ααρ }{ divides points on *
locX . Let }0{\*
locXu∈ , then ≠∈ )(|( tuStλ
0>)0≠ . Because of S is a subset of a real line which can be presented as no
more than numerable join of convex sets in R , we have :0 Θ∈∃α
0>)(
0
*
αI
u
X
. From here it follows 0>)(
0
uαρ , as it was to be shown.
From [12, theorem 1.37] it follows, that the given system of seminorms
Θ∈ααρ }{ generates some locally convex topology *X
τ on *
locX , which converts
the given space in locally convex linear topological space, whose local base we
obtain by find intersections of such sets:
{ }0>,|}<)(|{=),( *
loc εαερε αα Θ∈∈ uXuIV .
The statement b) follows from same theorem.
The lemma is proved.
Remark 5. Let us note that for every )(SBCI ∈ the space )(* IX is topo-
logically conjugated to )(IX , but *
locX is not topologically conjugated to locX .
For every 1≥n and )(SBCI ∈ we consider the Banach spaces
)();(=)(),();(=)( *
0
*
0
IXHILIXIXHILIX npnnqn ⊂⊂ ,
where },{max=: 210 rrp , 1=1
0
1
0
−− + pq with the natural norms. The space
);(
0 nq HIL is isometrically isomorphic to )(* IX n , the conjugate space of
)(IX n , moreover, the map
∫ ∫ 〉〈→∋×
I I
nXnHnn IxfdxfdxfxfIXIX )(,=))(),((=))(),((),()()( * ττττττ
is the duality form on )()( * IXIX nn × . This statement is correct in virtue of
);();();();();();( 22112100
∗∗
′′
+++⊂⊂ VILVILHILHILHILHIL qqrrqnq .
Let us point out that )(,=|,
)()( * InXIXIXI
nn
⋅〉〈⋅⋅〉〈⋅
×
.
Let us also consider the space
{ )}(||:=loc
αα
α IXyHSyX nInn ∈Θ∈∀→ ,
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 99
which topology is compatible with the set of seminorms ,})({ Θ∈⋅ ααInX and
{ })(||:= **
loc αα
α IXyHSyX nInn ∈Θ∈∀→ ,
which topology is compatible with the set of seminorms .})({ * Θ∈⋅ α
αInX
Proposition 1. For every 1≥n we have locloc = XPX nn , i.e.
{ }.)(|)(= locloc XffPX nn ∈⋅⋅
Moreover, if the triple ( )HVH ijj ;;}{ 1≥ , 1,2=i satisfies the condition (γ ) with
iCC = , then for each locXf ∈ , 1≥n and )(SBCI ∈ it results in ≤)(IXn fP
)(21 },{max IXfCC≤ .
Proof. To prove this proposition we will use [7] (proposition 3). Now we
consider the first part.
“⊂ ” Let loc
nXx∈ be arbitrary fixed. Then for almost all St∈
)(=)( txtxPn . Moreover, for every )(SBCI ∈ )()(| IXIXx nI ⊂∈ . Thus,
locXPx n∈ .
“⊃ ” Let locXPx n∈ be arbitrary fixed. Then for some locXy∈
)(=)( txtyPn for almost all St∈ . In virtue of [7](proposition 3) and the defini-
tion of locX it follows that for every )(SBCI ∈ )(|=| IXyPx nInI ∈ . Thus,
loc
nXx∈ .
The second part of the given proposition is the direct corollary of [7] (propo-
sition 3). This completes the proof.
Proposition 2. For every 1≥n we have *
loc
*
loc = XPX nn , i.e.
},)(|)({= *
loc
*
loc XyyPX nn ∈⋅⋅ and IIn yfyPf 〉〈〉〈 ,=, )(SBCI ∈∀ , *
locXy∈ ,
loc
nXf ∈ . Furthermore, if the triple ( )HVH ijj ;;}{ 1≥ , 1,2=i satisfies condition
(γ ) with iCC = , then it results in )(},{max)( *21* IyCCIyP
XXn ≤
)(SBCI ∈∀ , 1and*
loc ≥∈ nXy
Proof. To prove this proposition we use [7] (proposition 4). Now we con-
sider the first part.
“⊂ ” Let *
locnXf ∈ be arbitrary fixed. Then for almost all St∈
)(=)( tftfPn . Moreover, for every )(SBCI ∈ )()(| ** IXIXf nI ⊂∈ . Thus,
*
locXPf n∈ .
“⊃ ” Let *
locXPf n∈ be arbitrary fixed. Then for some *
locXg∈
( ) = ( )nP g t g t for almost all t S∈ . In virtue of [7](proposition 4) and the defini-
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 100
tion of *
locX it follows that for every )(SBCI ∈ )(|=| * IXfPf nInI ∈ . Thus,
*
locnXx∈ .
The last statements of the proposition is direct corollary of [7] (proposi-
tion 4).
Proposition is proved.
Proposition 3 Under the condition +∞<},{max 21 rr , the set *
loc
1
n
n
X∪
≥
is
dense in *
locX .
Proof. Arguing by contradiction, let us assume that for some *
locXf ∈ there
is an open set from the base of topology of the locally convex linear topological
space *
locX
),(=
1=
kk
n
k
V εα∩O ,
where 1≥n 0>kε , Θ∈kα , nk 1,= ,
0,>,,<)(|=),( *
*
loc εαεεα
α
Θ∈
⎭
⎬
⎫
⎩
⎨
⎧ ∈ IuXuV
X
such that
.=)(*
loc
1
∅+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≥
OfX n
n
∩∪
Thus
∅+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⊃+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≥≥
=)),(()( 00
*
loc
1
*
loc
1
εαVfXfX n
n
n
n
∩∪∩∪ O , (7)
where
k
nk
k
nk
IIBC
n αααεε ∪
1,=
00
1,=
0 )(:)(0,>min
1= ⊃∋Θ∈ RR . Because of the
set
( )
⎭⎬
⎫
⎩⎨
⎧ ∈++ ),(|||=|),( 00
000
00 εαεα
ααα
VggfVf III
is open in )(
0
*
αIX , due to [7](proposition 5) the set
)(=|
0
*
10
*
loc
1
αα
IXX n
n
In
n
∪∪
≥≥
⎟
⎠
⎞⎜
⎝
⎛ is dense in ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅ )(),(
0
*0
*
α
α IIX
X
and from (7) we obtain the contradiction.
The proof is concluded.
Now for an arbitrary )(SBCI ∈ we consider Banach space
)}(|)({=)( ** IXyIXyIW ∈′∈
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 101
with the norm ,)(*)(*)( IXIXIW
yyy ′+= where we mean the derivative y′
of an element *)(IXy ∈ in the sense of the scalar distribution space
));((=);( ***
wVIVI DLD , where *
wV is equal to *V with topology );( * VVσ
[10].
Together with *)(IW we consider the Banach space
1,2=)},(|);({=)(* iIXyVILyIW iipi ∈′∈ ,
with the norm
).();()(
*
)(* IWyyVIyIy iIXiipLiW
∈∀′+=
Also we consider the space )()(=)( *
2
*
1
*
0 IWIWIW ∩ with the norm
)();();(=)(
*
0)(2211
*
0
IWyyVIyVIyIy IXpLpLW
∈∀′++ .
Notice that the space )(* IW is continuously embedded in )(* IWi for
0,2=i .
Let us set },|);();({= loc
2
loc
21
loc
1
*
loc0 XyVSLVSLyW pp ∈′∩∈ where the
derivative y′ of an element );();( 2
loc
21
loc
1
VSLVSLy pp ∩∈ is regarded in the sense
of space of distributions );( ** VSD and in this space a subbase of topology σ is
assigned through the following sets:
⎩
⎨
⎧
⎩
⎨
⎧
++∈ );();(=),(=
2211
*
loc0 VIuVIuWuU
pLpL αα
εαC
}
⎭⎬
⎫Θ∈′+ 0>,<)( εαε
αIXu .
Lemma 3. C is a subbase of some topology σ in *
loc0W , which turns the
given space into separable locally convex linear topological space and, moreover:
a) σ is compatible with the set of seminorms
Θ∈′++ αααααρ }));();(=)({ (2211
IXpLpL uVIuVIuu ,
divides points on *
loc0W ;
b) a set *
loc0WE ⊂ is bounded only when for every Θ∈α αρ is bounded
on E .
Proof. As well as in lemma 1 it is enough to show that system of
seminorms Θ∈ααρ }{ divides points on *
loc0W . Let }0{\*
loc0Wu∈ , then
0>)0)(|( ≠∈ tuStλ . Because of S is a subset of a real line which can be
presented as no more than numerable join of convex sets in R , we have
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 102
Θ∈∃ 0α : 0>)(
0
*
0 αI
u
W
. From here it follows 0>)(
0
uαρ , as it was to be
shown.
From [12, theorem 1.37] it follows that the system of seminorms Θ∈ααρ }{
generates some locally convex topology σ on *
loc0W , which converts the given
space in locally convex linear topological space, whose local base we obtain by
final intersections of such sets:
{ }0>,|}<)(|{=),( *
loc0 εαερε αα Θ∈∈ uWuIV .
The statement b) follows from the same theorem.
The lemma is proved.
For a subset of a real line S which can be presented as no more than
numerable join of convex sets in R , distinct from a point, let us denote by
∆∈αα }{=)( ISBCC the family of all convex compact sets from S , distinct from a
point. We notice that the family of all subset of a real line S which can be
presented as no more than numerable join of convex sets in R , distinct from a
point, coincides with the family of all subset of a real line S which can be
presented as no more than numerable join of convex compact sets in R , distinct
from a point.
Let us also consider the space
{ )};(||:=);(loc HICyHSyHSC I αα
α ∈∆∈∀→
which topology is compatible with the set of seminorms ∆∈⋅ αα
});{ ( HIC .
Theorem 2. It results in );(loc*
loc0 HSCW ⊂ with continuous embedding.
Moreover, for every *
0, Wy ∈ξ and Sts ∈, : ts < and )(),( SBCts ∈ , the next
formula of integration by parts takes place
.))}(),(())(),({(=))(),(())(),(( ττξττξτξξ dyyssytty
t
s
′+′− ∫ (8)
In particular, when ξ=y we have:
τττ dyysyty
t
s
HH ))(),((=))()((
2
1 22 ′− ∫ .
Proof. At first let us prove the embedding );(loc*
loc0 HSCW ⊂ in the sense
of the set theory. Let *
loc0Wy∈ be fixed. Then for every St∈ , due to the set S
can be presented as no more than numerable join of convex compact sets in R ,
distinct from a point, there is )(SBCCI ∈ such that It ∈ . Moreover, we can
consider that t is an interior point of I in the space |)|,( ⋅S . Hence, due to the
definition of *
loc0W and [7, theorem 5] it follows that );()(| *
0 HICIWy I ⊂∈ .
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 103
Thus the function HSy →: is continuous in the point t . The necessary
statement follows from the arbitrary of St∈ .
Now let us prove the continuous embedding );(loc*
loc0 HSCW ⊂ . Since the
set S can be presented as no more than numerable join of convex compact sets in
R , distinct from a point, there exists ∆⊂Ξ ( 0card ℵ≤Ξ ) such that SI =α
α
∪
Ξ∈
.
So, it is enough to show that for every Ξ∈α there is a continuous seminorm
R→);(: loc HSCαµ and a constant 0>αC such that
*
loc0*
0
)()( WuuCIy
W
∈∀≤ αα
α
µ .
This fact follows from [7](theorem 5) because of for every Ξ∈α )(SBCCI ∈α .
At last we obtain formula (8) by using [7] (theorem 5) with ],[ tsS = .
The theorem is proved.
Let us consider the space },|{= loc**
loc XyXyW loc ∈′∈ which topology is
compatible with the set of seminorms Θ∈⋅ α
α
})({ * IW
.
In virtue of *
loc0
*
loc WW ⊂ with continuous embedding and due to the latter
theorem the next statement is true.
Corollary 1. );(loc*
loc HSCW ⊂ with continuous embedding. Moreover, for
every *
0, Wy ∈ξ and Sts ∈, : ts < and )(),( SBCts ∈ , formula (8) takes place.
For every 1n ≥ and ( )I BC S∈ let us introduce the Banach space
{ })(|)(=)( ** IXyIXyIW nnn ∈′∈
with the norm ,)()(=)( ** IyIyIy
nXnXnW
′+ where the derivative y′ is
considered in sense of scalar distributions space );(*
nHID and the space
}|{= loc*
loc
*
loc nnn XyXyW ∈′∈ ,
which topology is compatible with the set of seminorms Θ∈⋅ α
α
})({ * InW
.
As far as );(=));(());((=);( **** VSVSHSHS nn DDLDLD ω⊂ it is
possible to consider the derivative of an element )(* SXy n∈ in the sense of
);( ** VSD . Notice that for every 1≥n *
loc
*
loc1
*
loc WWW nn ⊂⊂ + .
Proposition 4. For every *
locXy∈ and 1≥n it results in )(= ′′ yPyP nn ,
where we mean the derivative of an element of *
locX in the sense of the scalar
distributions space );( ** VSD .
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 104
Remark 6. We point out that in virtue of the previous assumptions the de-
rivatives of an element of *
locnX in the sense of );( *VSD and in the sense of
);( nHSD coincide.
Proof. It is sufficient to show that for every )(SD∈ϕ )()(=)( ϕϕ ′′ yPyP nn .
In virtue of definition of derivative in sense of );( ** VSD we have
=)()(=)(=)()( ττϕτϕϕϕ dyPyPyPSD
S
nnn ′−′−′∈∀ ∫
).()(=))((=)()(= ϕϕττϕτ ′′−′−∫ yPyPdyP nnn
S
The proposition is proved.
From the propositions 2, 1, 4 it follows the next
Proposition 5. For every 1≥n *
loc
*
loc = WPW nn , i.e.
}.)(|)({= *
loc
*
loc lnn WyyPW ∈⋅⋅
Moreover, if the triple ( )HVH jii ;;}{ 1≥ , 1,2=j satisfies condition (γ ) with
jCC = , then for every *
locWy∈ , 1≥n and Θ∈α it results in
)()(},{max)()( *21*
αα IyCCIyP
WWn ⋅≤⋅ .
Theorem 3. Let the triple ( )HVH jii ;;}{ 1≥ , 1,2=j satisfy condition (γ )
with jCC = . Moreover, let *
locXD ⊂ be bounded in *
locX set and locXE ⊂
bounded in locX . For every 1≥n let us consider { ∈= nn yD :
} *
loc
*
loc and | nnnnn WEPyDyX ⊂∈′∈∈ Then for each Θ∈α , 1≥n and
nn Dy ∈
,)(*
αα
α
++ ⋅+≤ ECDIy
Wn (9)
where },{max= 21 CCC , )(sup= *
α
α
IyD
XDy∈
+ and )sup= ( α
α
IX
Ef
fE
∈
+ , i.e.
the set
*
loc
1
inboundedis WDn
n
∪
≥
and, consequently, bounded in );(loc HSC .
Remark 7. Due to proposition 1 nD is well-defined and *
locnn WD ⊂ .
Remark 8. A priori estimates (like (9)) appear at studying of global solvabil-
ity of differential–operator equations, inclusions and evolutional variational ine-
qualities in nonreflexive Banach and Frechet spaces with maps of λw -
pseudomonotone type by using Faedo–Galerkin method (see [1, 2]) at boundary
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 105
transition, when it is necessary to obtain a priori estimates of approximate solu-
tions ny in *
locX and its derivatives ny′ in locX .
Proof. The assertion of the theorem is immediate consequence of the ine-
quality
≤+∈Θ∈≥∀ ))(=)(,1, (
'
** ααα
α IXnXnWnnn yIyIyDyn
,},{max 21
αααα
++++ +≤+≤ ECCDEPD n
that is valid in virtue of proposition 1.
Further, let 0B , 1B , 2B be some Banach spaces such that
embeddingcompactwith,reflexiveare, 1020 BBBB ⊂ (10)
embeddingcontinuouswith210 BBB ⊂⊂ (11)
)[1;, 10 ∞+∈pp be arbitrary numbers. For every )(SBC∈α we consider the set
with the natural operations )};(|);({=)( 2100
BILvBILvIW pp ααα ∈′∈ , where
the derivative v′ of an element );( 00
BILv p α∈ is considered in the sense of the
scalar distribution space );( 2BIαD . It is obvious that ).;()( 00
BILIW p αα ⊂ Let
us also consider the set
)};(|);({= 2
loc
10
loc
0
loc BSLyBSLyW pp ∈′∈ .
It is clear, that ).;( 0
loc
0
loc BSLW p⊂
Theorem 4. locW with the natural operations, which is topologically com-
patible with set of seminorms Θ∈⋅⋅ αααρ })=)({ (IW is a Frechet space.
Proof. Since the set S can be presented as no more than numerable join of
convex sets in R there exists Θ⊂Ξ ( 0card ℵ≤Ξ ) such that
α
α
IS ∪
Ξ∈
= .
Thus, as well as in lemma 3, we can prove that the no more than numerable
system of seminorms Ξ∈⋅ ααρ )}({ divides points on locW . Thus, the families of
seminorms Ξ∈⋅ ααρ )}({ and Θ∈⋅ ααρ )}({ are equivalent and the locally convex
linear topological space ( )Ξ∈⋅ ααρ )}({,locW is metrizable.
Now let us prove that the metrizable space locW is complete. Let us consider
a Cauchy sequence loc
1}{ Wy nn ⊂≥ ; without loss of generality we can assume
that for every Ξ∈βα , : βα ≠ it follows that ∅∩ =βα II . We also consider
...}.<<<...<<{= 121 +Ξ nn αααα
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 106
)1i Because of loc
1}{ Wy nn ⊂≥ is a Cauchy sequence also
11
|
≥⎭⎬
⎫
⎩⎨
⎧
n
Iny
α
is a
Cauchy sequence in )(
1α
IW . Thus in virtue of [7] (theorem 8) there is a subse-
quence 11, }{ ≥nnv of 1}{ ≥nny that converges in )(
1α
IW to some )(
11 αIWx ∈ ;
2 )i analogously to )1i , due to loc
11, }{ Wv nn ⊂≥ is a Cauchy sequence the
same follows for
12
1, |
≥⎭⎬
⎫
⎩⎨
⎧
n
Inv
α
in )(
2α
IW . Thus there is a subsequence
12, }{ ≥nnv of 11, }{ ≥nnv that converges in )(
2α
IW to some )(
22 αIWx ∈ ;
)mi due to loc
1, }{ Wv nnm ⊂≥ is a Cauchy sequence the same follows for
1
, |
≥⎭⎬
⎫
⎩⎨
⎧
nm
Inmv
α
in ( )
m
W Iα . Thus there is a subsequence 1, 1{ }m n nv + ≥ of , 1{ }m n nv ≥
that converges in ( )
m
W Iα to some ( )m m
x W Iα∈ .
Thanks to 1 2), ),i i … , using the diagonal Cantor method, we can choose a
subsequence 1 , 1{ } = { }n k n n nk
y v≥ ≥ from 1{ }n ny ≥ that converges in ( )
m
W Iα to
)(
mm IWx α∈ for every 1≥m .
By setting )(=)( txty m ,
m
It α∈ , 1≥m we obtain that for every Ξ∈α
0)( →− yy
knαρ as ∞→k .
To conclude the proof we remark that locWy∈ in virtue of the definition
locW and the condition: Ξ∈∀ βα , : βα ≠ it follows that ∅∩ =βα II .
The theorem is proved.
Analogously with the proof of theorem 4 we can obtain the next:
Theorem 5. The set *
locW (respectively *
lociW , 0,2=i ) with the natural op-
erations, which topology is compatible with the set of seminorms Θ∈⋅ α
α
})({ * IW
(respectively Θ∈⋅ α
α
})({ * IiW
, 0,2=i ) is a Frechet space.
Theorem 6. Under conditions (10)–(11), we have );( 2
locloc BSCW ⊂ with
the continuous embedding.
Proof. At first let us prove the embedding );( 2
locloc BSCW ⊂ in the sense
of the set theory. Let locWy∈ be fixed. Then for every St∈ , since the set S can
be presented as no more than numerable join of convex compact sets in R , dis-
tinct from a point, there is )(SBCCI ∈ such that It ∈ . Moreover, we can con-
sider that t is an interior point of I in the space |)|,( ⋅S . Hence, due to the defini-
tion of locW and [7, theorem 5] it follows that );()(| 2BICIWy I ⊂∈ . Thus the
function 2: BSy → is continuous in the point t .
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 107
Now let us prove the continuous embedding );( 2
locloc BSCW ⊂ . Since the
set S can be presented as no more than numerable join of convex compact sets in
R , distinct from a point, there exists ∆⊂Ξ ( 0card ℵ≤Ξ ) such that SI =α
α
∪
Ξ∈
.
So, it is enough to show that for every Ξ∈α there is a continuous seminorm
R→);(: 2
loc BSCαµ and a constant 0>αC such that
loc
*
0
)()( WuuCIy
W
∈∀≤ αα
α
µ .
In fact for every Ξ∈α )(SBCCI ∈α . Thus the above inequality is true in virtue
of [7, theorem 9].
The theorem is proved.
The next result represents a generalization of the classical compactness
lemma [11] (theorem 1.5.1, p.70) into the case )[1;, 10 +∞∈pp .
Theorem 7. [7, theorem 10] Under conditions (10)–(11), for every
)[1;, 10 +∞∈pp the Banach space W is compactly embedded in );( 10
BSLp .
The proof follows from the next lemmas:
Lemma 4. [7, lemma 3] For every Wy∈ and R∈h it results in
,);();( 2121 BSyhBSyy LLh ′≤− where
⎩
⎨
⎧ ∈++
.otherwice,0
,),(
=)(
Shifthty
tyh
Lemma 5. [7, theorem 7] Let conditions (10)–(11) for 210 ,, BBB are valid,
)[1;, 10 +∞∈pp , S a finite time interval and );( 01
BSLK p⊂ such that
);;(inboundedis) 01
BSLKa p
b) for all 0>ε there exists 0>δ such that from δ<<0 h it results in
Kudphuu B
S
∈∀+−∫ ετττ <)()( 0
2
.
Then K is precompact in );(}; 110{min BSpL p .
Furthermore, if for some 1>q K is bounded in );( 1BSLq , then K is pre-
compact in );( 1BSLp for every )[1,qp∈ .
Lemma 6. [7, corollary 2] Let assumptions (10)–(11) for the Banach spaces
0B , 1B and 2B are valid, ][1;1 +∞∈p , ][0,TS = and the set );(
1
VSLK p⊂
such that
a) K is precompact set in );( 21
BSLp ;
b) K is bounded set in );( 01
BSLp .
Then K is precompact set in );( 11
BSLp .
P. Kasyanov, V. Mel'nik , A.-M. Piccirillo
ISSN 1681–6048 System Research & Information Technologies, 2007, № 4 108
The next result is a generalization of the compactness lemma [8] (theorem 2)
into the case )[1;, 10 +∞∈pp .
Theorem 8. Under above assumptions, the embedding locW in );( 1
loc
0
BSLp
is compact, that is, an arbitrary bounded in locW set is precompact in );( 1
loc
0
BSLp .
Proof. Arguing by contradiction, let loc
1}{ Wy nn ⊂≥ be bounded in locW
sequence that has no any accumulation point in );( 1
loc
0
BSLp . From [12, theorem
1.37] it follows, that for every convex bounded set SS ⊂α
+∞⎟
⎠
⎞
⎜
⎝
⎛ ′+
≥
<);();(sup
21001
BSyBSy
pLnpLn
n αα
. (12)
As on real line the arbitrary convex set can be presented as join no more,
than numerable number of bounded convex sets. Without loss of generality we
suppose α
α
SS ∪
Ξ∈
= , where αS is bounded convex set in R Ξ∈∀α and
0card ℵ≤Ξ . Further we consider only those Ξ∈α for which 0>)( αλ S .
Let it be 1}{= ≥Ξ nnα , then:
1)i from (12) and theorem 7 about compactness we obtain there is a subse-
quence 1, 1{ }n nv ≥ of 1{ }n ny ≥ , that is fundamental in the space 10 1
( ; )pL S Bα ;
2 )i analogously to 1)i , from (12) and theorem 7 about compactness it fol-
lows, there exists 2, 1 1, 1{ } { }n n n nv v≥ ≥⊂ that is fundamental in 10 2
( ; )pL S Bα ;
)mi from (12) and theorem 7 about compactness it follows, there exists
11,1, }{}{ ≥−≥ ⊂ nnmnnm vv , that is fundamental in );( 10
BSL
mp α ;
Thanks to …),), 21 ii , using the diagonal Cantor method, we can choose a
subsequence 1,1 }{=}{ ≥≥ nnnkkn vy from 1}{ ≥nny that is fundamental in
);( 1
loc
0
BSLp . This is a contradiction.
The theorem is proved.
By the analogy with the last theorem, due to the lemma 6, we can obtain the
next:
Theorem 9. Let assumptions (10)–(11) for the Banach spaces 0B , 1B and
2B are valid, )[1;1 +∞∈p , ][0,= TS and the set );(loc
1
VSLK p⊂ such that
a) K is precompact set in );( 2
loc
1
BSLp ;
b) K is bounded set in );( 0
loc
1
BSLp .
On some approximations and main topological descriptions for special classes of Frechet spaces …
Системні дослідження та інформаційні технології, 2007, № 4 109
Then K is precompact set in );( 1
loc
1
BSLp .
Now we combine all results to obtain the necessary a priori estimates.
Theorem 10. Let all conditions of theorem 3 are satisfied and HV ⊂ with
compact embedding. Then estimate (9) is true and the set
);(inprecompactand);(inboundedis locloc
1
HSLHSCD pn
n
∪
≥
for every 1≥p .
Proof. Estimation (9) follows from theorem 3. Now we apply the com-
pactness theorem 8 with 1=1,=,=,=,= 10
*
210 ppVBHBVB . Notice that
);(loc
1
*
loc VSLX ⊂ and );( *
1
loc VSLX ⊂ with continuous embedding. Hence, the
set
);(inprecompactis loc
1
1
HSLDn
n
∪
≥
.
In virtue of (9) and of theorem 2 on continuous embedding *
locW in
);(loc HSC it follows that the set
);(inboundedis loc
1
HSCDn
n
∪
≥
.
Further, we complete the proof by using standard conclusions, Lebesgue
theorem and the diagonal Cantor method.
Partially Supported by State Fund of Fundamental Investigations Grant
№ Ф25/539–2007.
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Received 03.08.2007
From the Editorial Board: the article corresponds completely to submitted
manuscript.
|
| id | journaliasakpiua-article-127326 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:23:33Z |
| publishDate | 2018 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/94/931d8bd119a10304c6d4dadbf791d294.pdf |
| spelling | journaliasakpiua-article-1273262018-04-11T11:11:24Z On some approximations and main topological descriptions for special classes of Frechet spaces with integrable derivatives О некоторых аппроксимациях и основных топологических характеристиках специальных классов пространств Фреше с интегрируемыми производными Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними Kasyanov, P. Mel'nik, V. Piccirillo, A.-M. We consider some classes of Frechet spaces with integrable derivatives. An important compactness lemma for nonreflexive spaces are obtained. Some main topological properties for the given spaces are obtained. Рассматриваются некоторые классы пространств Фреше с интегрированными производными. Сформулированы важные леммы о компактности для нерефлексированных пространств и их основные топологические особенности. Розглядаються деякі класи просторів Фреше з інтегрованими похідними. Сформульовано важливі леми про компактність для нерефлексованих просторів та їх основні топологічні властивості. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2018-03-29 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/127326 System research and information technologies; No. 4 (2007); 93-110 Системные исследования и информационные технологии; № 4 (2007); 93-110 Системні дослідження та інформаційні технології; № 4 (2007); 93-110 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/127326/122092 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Kasyanov, P. Mel'nik, V. Piccirillo, A.-M. Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними |
| title | Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними |
| title_alt | On some approximations and main topological descriptions for special classes of Frechet spaces with integrable derivatives О некоторых аппроксимациях и основных топологических характеристиках специальных классов пространств Фреше с интегрируемыми производными |
| title_full | Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними |
| title_fullStr | Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними |
| title_full_unstemmed | Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними |
| title_short | Про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів Фреше з інтегрованими похідними |
| title_sort | про деякі апроксимації та основні топологічні характеристики спеціальних класів просторів фреше з інтегрованими похідними |
| url | https://journal.iasa.kpi.ua/article/view/127326 |
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