Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1
We consider some classes of infinite-dimensional Banach spaces with integrable derivatives. An important compactness lemma for nonreflexive spaces is obtained. However some main topological properties for the given spaces are obtained.
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| Дата: | 2017 |
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2017
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Репозитарії
System research and information technologies| _version_ | 1867334323113295872 |
|---|---|
| author | Kasyanov, P. Mel’nik, V. |
| author_facet | Kasyanov, P. Mel’nik, V. |
| author_institution_txt_mv | [
{
"author": "P. Kasyanov",
"institution": null
},
{
"author": "V. Mel’nik",
"institution": null
}
] |
| author_sort | Kasyanov, P. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2018-04-11T11:06:06Z |
| description | We consider some classes of infinite-dimensional Banach spaces with integrable derivatives. An important compactness lemma for nonreflexive spaces is obtained. However some main topological properties for the given spaces are obtained. |
| first_indexed | 2025-07-17T10:23:11Z |
| format | Article |
| fulltext |
© P. Kasyanov, V. Mel'nik , 2008
Системні дослідження та інформаційні технології, 2008, № 1 127
UDC 517.9
ON SOME TOPOLOGICAL PROPERTIES FOR SPECIAL
CLASSES OF BANACH SPACES. PART 1
P. KASYANOV, V. MEL'NIK
We consider some classes of infinite-dimensional Banach spaces with integrable
derivatives. A compactness lemma for nonreflexive spaces is obtained. However
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
following: using the corresponding approximation scheme the approximate
solutions of a problem are constructed, for them some approaching a priori
estimations 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.
In the present paper we obtain a new of compact embedding theorems for
Banach spaces, suggested by researches about differential-operational inclusions
in function spaces. Moreover, we introduce some constructions to prove the con-
vergence of Faedo–Galerkin method for evolution variation inequalities with λw –
pseudomonotone maps [1–5].
In the following referring to Banach spaces YX , , when we write
YX ⊂
we mean the embedding in the set-theory sense and in the topological sense.
For 2≥n let n
iiX 1=}{ be some family of Banach spaces.
Definition 1. The interpolation family is refers a family of Banach spaces
n
iiX 1=}{ such that for some locally convex linear topological space (LTS) Y we
have
.1,=allfor niYX i ⊂
As 2=n the interpolation family is called the interpolation pair.
Further let n
iiX 1=}{ be some interpolation family. On the analogy of ([6], p.
23), in the linear variety i
n
i XX 1== ∩ we consider the norm
Xxxx
iX
n
i
X ∈∀∑
1=
=: , (1)
where
iX⋅ is the norm in iX .
Proposition 1. Let },,{ ZYX be an interpolation family. Then
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 128
XYYXZYXZYXZYX ∩∩∩∩∩∩∩∩ =,=)(=)(
both in the sense of equality of sets and in the sense of equality of norms.
We also consider the linear space
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
∈∑∑ niXxxXZ iii
n
i
i
n
i
1,=,==:
1=1=
with the norm
.=,maxinf=:
1=1,=
ZzzxXxxz i
n
i
iiiXi
niZ ∈∀
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
∈ ∑ (2)
Proposition 2. Let n
iiX 1=}{ be an interpolation family. Then i
n
i XX 1== ∩
and i
n
i XZ ∑ 1== are Banach spaces and it results in
.1,=allfor niZXX i ⊂⊂ (3)
Proof. Since X is a linear space, from properties of
iX⋅ and from the
definition of X⋅ on X it follows that X⋅ is the norm on X .
Let us prove the completeness of X . From the definition of X⋅ on X it
follows that every Cauchy sequence 1}{ ≥nnx in X is fundamental, so it
converges in iX and in Y ni 1,=∀ , where Y is the LTS in the definition 1.
Hence, due to n
iiX 1=}{ is the interpolation family and to the uniqueness of the
limit of a sequence 1}{ ≥nnx in LTS Y it follows that for some Xx∈ and for all
ni 1,=
.as ∞→→ nXinxx in
So, xxn → in X as ∞→n .
Now let us check that Z⋅ is the norm on Z .
If 0=Zz , then thanks to (2) for each 1≥m there exists imi Xx ∈ ( ni 1,= )
such that
n
xxz
iXmimi
n
i
1<,=
1=
∑ .
For every ni 1,= the sequence mix tends to 0 in iX , and so in Y too. Thus
01= →∑ mi
n
i x in Y as +∞→m and 0=z . On the other hand, let 0=z . Then
0=0max
1,= iXniZz ≤ .
The another norm properties for Z⋅ follow from the properties of inf ,
max and norms
iX⋅ , ni 1,= .
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 129
Let us check Z under the above norm is complete space. Let 1}{ ≥mmz be a
Cauchy sequence in Z . It contains a subsequence 1}{ ≥kkmz with the property
2.for2<
1
≥− −
−
kzz k
Zkmkm
From (2) for every 2≥k there exists
,=
1=
1 kj
n
j
kmkm uzz ∑−
−
where jkj Xu ∈ , k
Xkju −12< for each nj 1,= and 2≥k . Further,
njXuuz jj
n
j
jm ,1,, 1
1
11
=∈=∑
=
.
For every 1≥k let us put
.1,=,=
1=
njux ij
k
i
kj ∑
Hence
1.=
1=
≥∀∑ kxz kj
n
j
km
For all nj 1,= the sequence kjx converges in jX (according to its construction)
to some jj Xx ∈ . Let us set j
n
j
xz ∑
1=
= . Then we have
1.max
1,=
≥∀−≤− kxxzz
jXkjj
njZkm
From here it follows that
kmz converges to z in Z as +∞→k . From the
estimation
ZmkmZkmZm zzzzzz −+−≤−
and taking into account that the sequence 1}{ ≥mmz is fundamental we obtain
0.=lim Zmm
zz −
∞→
The embedding (3) follows from the definition of Banach spaces ),( XX ⋅
and ),( ZZ ⋅ .
Remark 1. ([6], p. 24). Let Banach spaces X and Y satisfy the following
conditions
.const=,
,indenseis,
γγ Xxxx
YXYX
XY ∈∀≤
⊂
Then
., *
**
** YfffXY YX ∈∀≤⊂ γ
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 130
Moreover, if X is reflexive, then *Y is dense in *X .
Let n
iiX 1=}{ be an interpolation family such that the space i
n
i XX 1==: ∩ with
the norm (1) is dense in iX for all ni 1,= . Due to remark 4 the space *
iX may be
considered as subspace of *X . Thus we can construct *
1= i
n
i X∑ and
.
*
1
*
1=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⊂
=
∑ i
n
i
i
n
i
XX ∩ (4)
Under the given assumptions X is dense in i
n
i
XZ ∑
1=
=: for every ni 1,= . So
iX is dense in Z too. Thanks to remark 1 we can consider space *Z as a
subspace of *
iX for all ni 1,= , and also as a subspace of *
1= i
n
i X∩ , i.e.
.*
1
*
1=
i
n
i
i
n
i
XX
=
⊂⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∑ ∩ (5)
Theorem 1. Let n
iiX 1=}{ be an interpolation family such that the space
i
n
i
XX
1
=:
=
∩ with the norm (1) is dense in iX for all ni 1,= . Then
*
1
*
1=
= ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
∑ i
n
i
i
n
i
XX ∩ and *
1
*
1=
= i
n
i
i
n
i
XX
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∑ ∩
both in the sense of sets equality and in the sense of the equality of norms.
Proof. We consider the space i
n
i
X∏ 1=
=:X with the norm
XX ∈∀∑ },...,,{==},...,,{ 21
1=
21 niXi
n
i
n xxxxxxxx ;
let L be the subspace of X defined by
= {{ , ,..., }| }.x x x x X∈L
For a fixed *Xf ∈ let us set
.)(=}),...,,({ Xxxfxxxu ∈∀
Hence u is a linear functional on L with the norm .= ** Xfu By Hahn–
Banach theorem for the functional u there exists a linear functional v defined on
X such that
.== ** Xfuv
For every ni 1,= we set
iiii Xxxvxg ∈∀})0,...,0,,0,...,0({=)( .
Hence it is clear that *
ii Xg ∈ for all ni 1,= and
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 131
***
1,=
=max XZiXi
ni
fvg ≤ .
By the construction,
,)(=)(
1=
Xxxgxf i
n
i
∈∀∑
i.e. *
1=1== i
n
ii
n
i Xgf ∑∑ ∈ . Thus it follows
.max **
1,=
*
1
XiXi
nii
n
i
fgXf ≤≤
=Σ
On the other hand
≤
∑
)(sup
1=
=
1=
* xff
iXf
n
i
X
≤
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
=∈≤ ∑∑
===∑
=
n
i
iiiX
n
i
Xi
x
fgXgxg
iin
i iX
1
*
11
,infsup *
1
∑ =⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
∈≤ ∑ n
i iXi
n
i
iiiX
ni
ffgXg
1
*==,gmaxinf
1=
*
*i
1,=
.
The latest inequalities and (4) prove the first part of the theorem.
Let us prove the remaining part.
Lemma 1. Let *
1= i
n
i Xf ∩∈ . Then for every nk 2,= and iii Xyx ∈,
( = 1,i k ) such that xyx i
k
ii
k
i :== 1=1= ∑∑ we have
).(:=)(=)(
1=1=
xfyfxf i
k
i
i
k
i
∑∑ (6)
Proof. We prove this statement arguing by induction.
Let iii Xyx ∈, ( 1,2=i ) such that xyyxx :== 2121 ++ . Then =− 11 yx
2122 XXxy ∩∈−= and
).()(=)(=)(=)()( 22221111 xfyfxyfyxfyfxf −−−−
From the last the necessary statement is follows.
Now we assume that for some 12,= −nk and for arbitrary iii Xyx ∈,
( ki 1,= ) such that xyx i
k
i
i
k
i
:
1=1=
==∑∑ equality (6) is valid.
Let iii Xyx ∈, ( 11, += ki ) such that xyx i
k
i
i
k
i
:
1
1=
1
1=
==∑∑
++
. Thus
,)(= 1
1=1=
11 +++ ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∈−− ∑∑ ki
k
i
ii
k
i
kk XXxyyx ∩
and so, by the induction assumption, we obtain
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 132
))()((=)(=)(=)()(
1=1=
1111 ii
k
i
ii
k
i
kkkk xfyfxyfyxfyfxf −⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−− ∑∑++++
and the lemma follows.
According to lemma 1 let us continues any fixed functional *
1= i
n
i Xf ∩∈ to
some functional on Z in such way:
for i
n
i xz ∑ 1== , where ii Xx ∈ ni 1,=∀ ,
).(=)(
1=
i
n
i
xfzf ∑
From relation (6) it follows that the given definition is correct and does not
depend on the representation of z as i
n
i x∑ 1= . Since
ZiX
n
i
i
n
i
iiiXiiX
n
i
zfzxXxxfzf ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≤
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
∈≤ ∑∑∑ *
1=1=
*
1=
=,inf)( ,
then *Zf ∈ and .*
1=
*
i
n
iZ Xff ∩≤ Taking into account (5) we have =*Z
*
1= i
n
i X∩= as equality of the sets. In order to prove the equality of norms it is
sufficient to show the inequality **
1= Zi
n
i
fXf ≤∩ . For every 0>ε there exists
ii Xx ∈ such that
1=,/)(*
iXiiiX xnxff ε+≤ .
Hence
≤+≤+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≤ ∑∑∑∩ εε
Z
i
n
i
Zi
n
iiX
n
ii
n
i
xfxffXf
1=
*
1=
*
1=
*
1=
=
εε ++≤ *
1,=
* =max ZiX
ni
Z fff
and from here the delivered conclusion follows.
Now let Y be some Banach space, *Y its topological conjugated space, S
be some compact time interval. We consider the classes of functions defined on
S and imagines in Y (or in *Y ).
The set );( YSLp of all measured by Bochner functions [6] as +∞≤≤ p1
with the natural linear operations with the norm
p
p
Y
S
pL dttyYSy
1/
)(=);( ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∫
is a Banach space. As +∞=p );( YSL∞ with the norm
Y
St
L tyYSy )(maxvrai=);(
∈∞
is a Banach space.
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 133
The next theorem shows that under the assumption of reflexivity or
separability of Y the conjugated to );( YSLp , +∞≤ <1 p , space *));(( YSLp may
be identify with );( *YSLq , where q is such that 1 1 = 1p q− −+ .
Theorem 2. If the space Y is reflexive and +∞≤ <1 p , then each element
*));(( YSLf p∈ has the unique representation
);(everyfor)(),(=)( YSLydttytyf pY
S
∈〉〈∫ ξ
with the function );( *YSLq∈ξ , 1=11 −− + qp . The correspondence ξ→f , with
*));(( YSLf p∈ is linear and
.);(=));( **( YSYSf
qLpL 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 for the given
spaces let the next conditions be satisfied
VHV ,⊂ is dense in H ,
Vvvv VH ∈∀≤>∃ :0γ . (7)
Due to remark 1 under the given assumptions we may consider the
conjugated to H space *H as a subspace of *V that is conjugated to V . As V
is reflexive then *H is dense in *V and
,*
** Hfff HV ∈∀≤ γ
where *V⋅ and *H⋅ are the norm in *V and in *H , respectively.
Further, we identify the spaces H and *H . Then we obtain
V H V ∗⊂ ⊂
with continuous and dense embedding.
Definition 2. The triple of spaces ( *;; VHV ), that satisfy the latter conditions
will be called the evolution triple.
Let us point out that under identification H with *H and *H with some
subspace of *V , an element Hy∈ is identified with some *Vf y ∈ and
,,=),( Vxxfxy Vy ∈∀〉〈
where V⋅〉〈⋅, is the canonical pairing between *V and V . Since the element y
and yf are identified then, under condition (7), the pairing , V〈⋅ ⋅〉 will denote the
inner product on H ( , )⋅ ⋅ .
We consider ip , ir , 21,=i such that +∞≤≤ ii rp<1 , +∞<ip . Let
1≥≥ ′ii rq well-defined by
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 134
1,2=1== 1111 irrqp iiii ∀++ −
′
−−− .
Remark that 0=1/∞ .
Now we consider some Banach spaces that play an important role in the
investigation the differential-operator equations and evolution variational
inequalities in non-reflexive Banach spaces.
Referring to evolution triples ( *;; ii VHV ) ( 1,2=i ) such that
HVVVV and,spacestheindenseissetthe 2121 ∩ (8)
we consider the functional Banach spaces (proposition 2)
1,2=),;();(=)(= * iHSLVSLSXX
iriiqii ′
+
with norms
⎪⎩
⎪
⎨
⎧
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
);(;);(maxinf=
'
2*1 HSyVSyy
ir
LiiqLiX
⎭
⎬
⎫+∈∈
′ 212
*
1 =),;(),;( yyyHSLyVSLy
iriiq ,
for all iXy∈ , and
* *
1 21 2 2 1
= ( ) = ( ; ) ( ; ) ( ; ) ( ; )q q r rX X S L S V L S V L S H L S H
′ ′
+ + +
with
);(|});(;);({max{inf= *
12*1
1,2=
iiqi
ir
LiiiqLi
iX VSLyHSyVSyy ∈
′
,
}=1,2;=),;( 222112112 yyyyyiHSLy
iri +++∈
′
,
for each Xy∈ . As +∞<ir , due to theorem 1 and to theorem 2 the space iX is
reflexive. Analogously, if +∞<},{max 21 rr , the space X is reflexive.
Under the latter theorems we identify the conjugated to )(SX i , )(= ** SXX ii ,
with );();( iipir
VSLHSL ∩ , where
*
* );();(= iiipL
ir
LiX XyVSyHSyy ∈∀+ ,
and, respectively, the conjugated to )(SX space )(= ** SXX we identify with
);();();();( 221121
VSLVSLHSLHSL pprr ∩∩∩ ,
where
*
221121
* );();();();(=)( XyVSyVSyHSyHSySy
pLpLrLrLX ∈∀+++ .
On )()( * SXSX × we denote by
++〉〈〉〈 ∫∫ ττττττ dyfdyfyfyf H
S
H
S
S ))(),(())(),((=,=, 1211
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 135
=)(),()(),(
222121 ττττττ dyfdyf V
S
V
S
〉〈+〉〈+ ∫∫
*,))(),((= XXfdyf
S
∈∈∀∫ τττ ,
where 22211211= fffff +++ , );(1 HSLf
iri ′
∈ , );( *
2 iiqi VSLf ∈ , 1,2=i .
Let 21= VVV ∩ , ( )VF be a filter of all finite-dimensional subspaces from
V . As V is separable, there exists countable monotone increasing system of
subspaces 1{ } ( )i iH V≥ ⊂ 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 orthogonal projection from H on nH operator:
.minarg=everyfor Hn
nnh
n hh
H
hPHh −
∈
∈
Definition 3. We say that the triple ( )HVH ii ;;}{ 1≥ satisfies condition (γ ), if
+∞
≥
< sup ),(
1
VVLn
n
P , i.e. there exists such 1≥C that for every Vv∈ and 1≥n
VVn vCvP ≤ . (9)
Some constructions that satisfy condition (γ ) were presented in [7].
Remark 2. It is easy to notice that there exists such complete orthonormal in
H vector system Vh ii ⊂≥1}{ that for any 1≥n nH is a linear capsule stretched
on n
iih 1=}{ . Then condition ( )γ means that the given system is a Schauder basis
in the space V ([8], p. 403).
Remark 3. Due to the identification of *H and H it follows that *
nH and
nH are identified too.
Remark 4. Since ),( VVLnn PP ∈ for every 1≥n we get
),(
**
** VVLnn PP ∈
and
),(
*
),( **=
VVLnVVn PP L . It is clear that for every Hh∈ hPhP nn
*= . Hence,
we identify nP with its conjugate *
nP for every 1≥n . Then, condition )(γ means
that for every Vv∈ and 1≥n it results in
.and ** VVnVVn vCvPvCvP ≤≤ (10)
For each 1≥n we consider the Banach spaces
,);(=)(=,);(=)(= *
0
**
0
XHSLSXXXHSLSXX npnnnqnn ⊂⊂
where },{max=: 210 rrp , 1=1
0
1
0
−− + pq with the natural norms. The space
);(
0 nq HSL is isometrically isomorphic to the conjugate space *
nX of nX ,
moreover, the map
nXSnHSnn xfdxfdxfxfXX 〉〈→∋× ∫∫ ,=))(),((=))(),((,* ττττττ
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 136
is the duality form on *
nn XX × . This statement is correct due to
);();();();();();( 22112100
∗∗
′′
+++⊂⊂ VSLVSLHSLHSLHSLHSL qqrrqnq .
Let us remark that )(,=)()(|, * SSXS nXnnXS ⋅〉〈⋅×⋅〉〈⋅ .
Proposition 3. For every 1≥n XPX nn = , i.e.
{ }XffPX nn ∈⋅⋅ )(|)(= .
Moreover, if the triple ( )HVH ijj ;;}{ 1≥ , 1,2=i satisfies condition (γ ) with
iCC = , then
XXn fCCPnXf },{max1andeveryfor 21≤≥∈ .
Proof. Let us fix an arbitrary number 1≥n . For every Xy∈ let
)(:=)( ⋅⋅ yPy nn , i.e. )(=)( tyPty nn for almost all St∈ . Since nP is linear and
continuous on *
1V , on *
2V and on H we have that XXy nn ⊂∈ . It is immediate
that the inverse inclusion is valid.
Now let us prove the second part of this statement. We suppose that
condition (γ ) holds, let us fix Xf ∈ and 1≥n . Then from condition (γ ) it
follows that for every );(1 HSLf
iri ′
∈ and );( *
2 iiqi VSLf ∈ such that
22211211= fffff +++ we have
+++
′′
);();();( *
11
21
2
12
1
11 VSfPHSfPHSfP
qLnrLnrLn
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+ ′′ ∫∫
'
2
'
1
1
2
12
1
1
11*
22
22 )()(=);(
rr
Hn
S
rr
Hn
S
qLn dfPdfPVSfP ττττ
≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+ ∫∫
2
1
2
*
2
22
1
1
1
*
1
21 )()(
q
Vn
S
q
Vn
S
dqfPdqfP ττττ
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≤
′
′
′
′ ∫∫
2
1
2
12
1
1
1
11 )()(
rr
H
S
rr
H
S
dfdf ττττ
≤
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+ ∫∫
2
1
*
2
222
1
*
1
211
211 )()(
qq
V
S
qq
V
S
dfCdfC ττττ
⎜
⎝
⎛ ++≤
′′
);();(},{max
2
12
1
1121 HSfHSfCC
rLrL
⎟⎟
⎠
⎞
++ );();( *
22
22*
11
21 VSfVSf
qLqL
,
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 137
because 1, 21 ≥CC . Therefore, due to the definition of norm in X we complete
the proof.
Proposition 4. For every 1≥n it results in ** = XPX nn , i.e.
},)(|)({= ** XyyPX nn ∈⋅⋅
and
.and,=, *
nn XfXyyfyPf ∈∈∀〉〈〉〈
Furthermore, if the triple ( )HVH ijj ;;}{ 1≥ , 1,2=i satisfies condition (γ ) with
iCC = , then we get
1and},{max *
*21* ≥∈∀≤ nXyyCCP XXn .
Proof. For every *Xy∈ we set )(:=)( ⋅⋅ yPy nn , i.e. )(=)( tyPty nn for a.e.
St∈ . As the operator nP is linear and continuous on 1V , on 2V and on H we
have that ** XXy nn ⊂∈ . The inverse inclusion is obvious.
Due to condition (γ ) and to the definition of );( iipL VS⋅ and );( HS
ir
L⋅
it follows that
.);();(and);();( HSyHSyVSyCVSy
ir
L
ir
LniipLiiipLn ≤⋅≤
Thus *21* },{max XXn yCCy ≤ .
Now let us show that for every nXf ∈
.,=, 〉〈〉〈 yfyf n
As );(
0 np HSLf ∈ , then we have
=))(),((=))(),((=, ττττττ dyPfdyfyf nSS ∫∫〉〈
,,=))(),((= 〉〈∫ nnS
yfdyf τττ
because for every 1≥n , Hh∈ and nv H∈ it results in
0=),(=),( Hnn vhPhvhPh −− .
The proposition is proved.
Proposition 5. Under the condition +∞<},{max 21 rr the set *
1
n
n
X∪
≥
is dense
in ),( *
*
XX ⋅ .
Proof. a) At first we prove that the set );( VSL∞ is dense in space
),( *
*
XX ⋅ .
Let us fix *Xx∈ .
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 138
Then for every 1≥n we consider
⎪⎩
⎪
⎨
⎧ ≤
=
.elsewhere,0
,)()(
:)(
ntxtx
tx V
n (11)
Obviously );( VSLxn ∞∈ . The continuous embedding of V into H assures
the existence of some positive γ such that for 1,2=i and a.e. St∈ we have
⎪⎭
⎪
⎬
⎫
∞→→−≤−
→−≤−
,as,0)()()()(
,0)()()()(
ntxtxtxtx
txtxtxtx
VnVn
VnHn
i
γ
(12)
.)()(,)()(
iViVnHHn txtxtxtx ≤≤ (13)
Further let us set
.)()(=)(,)()(=)( 0 ip
iVniV
p
Hn
n
H txtxtntxtxt −− φφ
So, from (12) and (13) we obtain
Stntnt
iV
n
H ∈∞→→→ .a.eforas0)(0,)( φφ (14)
and for almost every St∈
)(:=)(2|)(|),(:=)(2|)(| 00 tptxtnttxt
iV
i
iV
ip
iVH
p
H
pn
H φφφφ ≤≤ . (15)
Since *Xx∈ , then )(,, 121
SLVVH ∈φφφ . Thus, due to (14) and (15), we can
apply the Lebesgue theorem with integrable majorants Hφ ,
1Vφ and
2Vφ
respectively. Hence it follows that 0→n
Hφ and 0→n
iVφ in )(1 SL as 1,2=i .
Consequently 0* →− Xn xx as ∞→n .
b) Further, for some linear variety L from V we set
}functionsimpleais|)({=:)( yLSyL →∈ϒ
([6], p.152). Let us show that set )(Vϒ is dense in the normalized space
)),,(( *XVSL ⋅∞ . Let be x fixed in ),( VSL∞ ; so, there exists a sequence
)(}{ 1 Vx nn ϒ⊂≥ such that
StnVtxtxn ∈∞→→ a.e.forasin)()( . (16)
Since ),( VSLx ∞∈ we have +∞
∈
<:=)(supess ctx V
St
. For every 1≥n let us
introduce
⎪⎩
⎪
⎨
⎧ ≤
=
.else,0
,2)(),(
:)(
ctxtx
ty Vnn
n (17)
From (16) and (17) it follows that )(Vyn ϒ∈ as 1≥n and moreover,
StnVtxtyn ∈∞→→ a.e.forasin)()( .
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 139
Hence, taking into account HV ⊂ , as 1,2=i and for a.e. St∈ we obtain the
following convergences
∞→→→→ nVtxtyVtxtyHtxty nnn asin)()(,in)()(,in)()( 21 .
As in a), assuming
))(})(3,)(3,){(3max 1
021
21
SLccc ppp
VVH ∈≡≡≡ γφφφ
we obtain that xyn → in *X as ∞→n . So, )(Vϒ is dense in
).),,(( *XVSL ⋅∞
c) Since the set n
n
nn Hh
1
1 =}{span
≥
≥ ∪ is dense in ),( VV ⋅ and HV ⊂ with
continuous embedding it follows that the set
)(=
11
n
n
n
n
HH ϒ⎟
⎠
⎞
⎜
⎝
⎛ϒ
≥≥
∪∪ is dense in )),(( *XV ⋅ϒ .
In order to complete the proof we point out that for every 1≥n
nn XH *)( ⊂ϒ . The proposition is proved.
Now we consider Banach space }|{= ** XyXyW ∈′∈ with the norm
,= ** XXW yyy ′+
where the derivative y′ of an element *Xy∈ is in the sense of the scalar
distribution space * * *( ; ) = ( ( ); )wS V S VD L D , where *
wV be equals to *V with
topology );( * VVσ [9].
Together with )(= ** SWW we consider the Banach space
1,2,=)},(|);({=)(= ** iSXyVSLySWW iipii ∈′∈
with the norm
.);(= *
* iXiipLiW WyyVSyy ∈∀′+
We also consider the space )()(=)(= *
2
*
1
*
0
*
0 SWSWSWW ∩ with the norm
.);();(= *
02211
*
0
WyyVSyVSyy XpLpLW ∈∀′++
The space *W is continuously embedded in *
iW for 0,2=i .
Theorem 3. It results in );( ** VSCWi ⊂ with continuous embedding for
0,2=i .
Proof. Let 1,2=i be fixed, *
iWy∈ and Stt ∈∀ ,0 we set ττξ dy
t
t
t
)()(
0
′= ∫
which has sense in the virtue of the local integrability of y′ . It is obvious that
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 140
tsdyst V
s
t
V ≥∀′≤− ∫ ττξξ ** )()()(
from which follows );( *VSC∈ξ . Then y′′ =ξ , it means that ztty += )()( ξ for
a.e. St∈ and some *Vz∈ . Therefore, the function y also belongs to );( *VSC .
Note, that S is compact. Then in virtue of );( *
1 VSLX ⊂ we obtain
Stykdyt XV
S
V ∈∀′≤′≤ ∫ ττξ ** )()( .
Then due to the continuity of embedding *VVi ⊂ we have
≤−∫ )*;(
1/
*
1/
* =)(=))(mes( VS
ipL
ipi
V
S
ip
V ydspzSz ξ
));(()));(( 2*;(*1 XiipLVSC
ipL yVSykVSyk ′+≤+≤ ξ (18)
where 2k does not depend on *
iWy∈ .
Now let );( **
0 VSCWy ⊂∈ . In virtue of (18) for 1,2=i there exists 03 ≥k
that
03*;( ) WVSC yky ≤ for all *
0Wy∈ .
Remark 5. From the definition of norms in the spaces *W and *
0W we
obtain );( ** VSCW ⊂ with continuous embedding for the compact S in the
natural topology of the space *W .
Theorem 4. The set *
0
1 );( WVSC ∩ is dense in *
0W .
Proof. We prove this statement for more general case. At the beginning we
suppose R=S . Let us choose such a function )(0 SCK ∞∈ that 1=)( ττ dK
S
∫ and
use the Sobolev mid-value method. Let us set for definiteness
⎪⎩
⎪
⎨
⎧
≤
−
−
1,|>|for0
1,||for}
1
{exp=)( 2
2
τ
τ
τ
τµτK
where µ is the constant of normalization and suppose )(=)( ττ nnKKn for every
S∈τ and 1≥n . It is obvious that )(0 SCKn
∞∈ and
11=)( ≥∀∫ ndKn
S
ττ .
For every *
0Wy∈ let us define the sequence of functions
.)()(=)( τττ dytKty n
S
n −∫ (19)
It is easy to check that );(1 VSCyn ∈ and
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 141
.)()(=)()(= ττττττ dytKdytKy n
S
n
S
n ′−−′′ ∫∫ (20)
Besides );( iipn VSLy ∈ and yyn → in );( iip VSL for ( 1,2=i ). The last
follows from the inequality );()();( 1 iipLLiipLn VSySKVSy ≤ and from fol-
lowing estimations:
≤−−− ∫∫
+
−
dtpdtyytKp
VSyy i
iVn
nt
ntS
i
iipLn τττ ))()()((=);(
1/
1/
≤−+≤ ∫∫∫
−−
dtdsptystyqdssK i
iV
n
n
iipiq
n
n
nS
})()(/)|)(|{(
1/
1/
1/
1/
dsdtptystyn i
iV
S
n
n
ip ))()(()(2
2
1/
1/
−+≤ ∫∫
−
µ .
Pointing out that for arbitrary )<(1);( ∞≤∈ iiip pVSLy and for every h
the function
⎩
⎨
⎧
∈/+
∈++
Sht
Shthty
tyh for0
,for)(
=)(
belongs to );( iip VSL and 0);( →−
iipLh VSyy as 0→h [6, lemma IV.1.5],
then
1,2=for0=);(sup)(2lim);(lim
1/||
ip
VSyyp
VSyy i
iipLs
ns
ip
n
i
iipLn
n
−≤−
≤∞→∞→
µ .
Now we prove the convergence of derivatives. Let Xy ∈′ and =′y
2121 ηηξξ +++= where );( *
iiqi VSL∈ξ , );( HSL
iri ′
∈η , 1,2=i . By the anal-
ogy with (19) we suppose
∫∫ −=−=
S
inin
S
inin dtKtdtKt ττητηττξτξ )()()(,)()()( ,, for 2,1=i .
Then in virtue of (20) by the analogy to the previous case, +′ ,1= nny ξ
,2,1,2 nnn ηηξ +++ and besides iin ξξ →, in );( *
iiq VSL and iin ηη →, in
);( HSL
ir ′
for 1,2=i . By definition of X⋅ , it follows
⎩
⎨
⎧
−−≤′−′
∞→∞→ );(;);(maxlimlim *
22
,2*
11
,1 VSVSyy
qLn
qLnnXnn
ξξξξ ;
0=);(;);(
2
,2
1
,1
⎭
⎬
⎫−−
′′
HSHS rLnrLn ηηηη
From here we conclude that for every 1≥n *
0
1 );( WVSCyn ∩∈ and the se-
quence 1}{ ≥nny converges to *
0Wy∈ in *
0W .
P. Kasyanov, V. Mel'nik
ISSN 1681–6048 System Research & Information Technologies, 2008, № 1 142
Now let us consider the case when S is semi-bounded. Without loss of gen-
erality we suppose )[0,= ∞S . For )(= *
0
*
0 SWWy ∈ we put )(=)( htytyh + for
every 0>h . Then, in virtue of [6, lemma IV.1.5] it is easy to show that for
1,2=i yyh → in );( iip VSL and yyh ′→′ in X as +→ 0h . Remark that
*
0Wyh ∈ . To complete the proof it is sufficient to show that for every fixed
)(*
0 SWy∈ and for 0>h the element *
0Wyh ∈ can be sufficiently exactly ap-
proximated by the functions from *
0
1 );( WVSC ∩ .
For some *
0 ( )y W S∈ and > 0h let us consider the function
⎩
⎨
⎧
−
−≥+
,<for0
,for)()(
=)(
ht
hthtyt
t
ϕ
ξ
where )(1 RC∈ϕ , 1=)(tϕ if
2
ht −≥ and 0=)(tϕ if ht −< . Then for every
0≥t )(=)( tyt hξ and due to definition of derivative in sense of scalar distribu-
tion space );( ** VSD it follows that
⎩
⎨
⎧
−
−≥+′++′
′
.<for0
,for)()()()(
=)(
ht
hthtythtyt
t
ϕϕ
ξ
Let us prove that )(*
0 RW∈ξ . Since )(*
0 SWyh ∈ we have )(| *
)[0, SX∈∞ξ .
Because of 0=| );( h−−∞ξ it remains to consider the section ,0)[ h− .
From 1|=)(|sup
,0)[
s
hs
ϕ
−∈
we have
≤+≤ ∫∫
−−
dsphsysdsps i
iV
ip
h
i
iV
h
)(|)(|)(
00
ϕξ
.1,2)=()(=)(
0
0
idpydsphsy i
iV
h
i
iV
h
ττ∫∫ +≤
−
Thus, ),0);([| ,0)[ iiph VhL −∈−ξ for 1,2=i . Similarly we can prove that
)(RX∈′ξ . So, )(*
0 RW∈ξ and in virtue of the previous case there exists a se-
quence of elements )();( *
0
1 RR WVCn ∩∈ξ that converges to ξ in )(*
0 RW .
Now we set )();(|= *
0
1 SWVSCSnn ∩∈ξζ for every 1≥n . Here hn y→ζ
in )(*
0 SW as ∞→n , because hS y=|ξ .
Let us consider, at last, the case when S is bounded. For every )(*
0 SWy ∈
(where ],[= baS , ba < ) we put
⎩
⎨
⎧ ∈
,>for0
],,[for)()(
=)(
bt
battyt
t
ϕ
ξ
On some topological properties for special classes of Banach spaces. Part 1
Системні дослідження та інформаційні технології, 2008, № 1 143
⎩
⎨
⎧ ∈−
.<for0
],,[for)())((1
=)(
at
battyt
t
ϕ
η
Let ϕ be such function from )(1 SC that 0=)(tϕ in some neighborhood of
the point b and 1=)(tϕ in some neighborhood of the point a . Note that
)()(=)( ttty ηξ + for all St∈ . It is easy to check that )),([*
0 ∞∈ aWξ and
]),((*
0 bW −∞∈η . Therefore, due to the previous case, there exist such sequences
)),([));,([}{ *
0
1
1 ∞∩∞⊂≥ aWVaCnnξ
and
1 *
1 0{ } (( , ); ) (( , )),n n C b V W bη ≥ ⊂ −∞ ∩ −∞
that
∞→−∞→∞→ nbWaW nn as)),((inand)),([in *
0
*
0 ηηξξ .
So, ySnn →+ |)( ηξ in )(*
0 SW .
The theorem is proved.
Partially Supported by State Fund of Fundamntal Investigations Grant
№ Ф25.1/029
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Received 05.07.2007
From the Editorial Board: the article corresponds completely to submitted
manuscript.
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| id | journaliasakpiua-article-109780 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:23:11Z |
| publishDate | 2017 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/f5/b420179c8502b9108cb488e8cbfd7ef5.pdf |
| spelling | journaliasakpiua-article-1097802018-04-11T11:06:06Z On some topological properties for special classes of Banach spaces. Part 1 О некоторых топологических свойствах специальных классов банаховых пространств. Часть 1 Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1 Kasyanov, P. Mel’nik, V. We consider some classes of infinite-dimensional Banach spaces with integrable derivatives. An important compactness lemma for nonreflexive spaces is obtained. However some main topological properties for the given spaces are obtained. Рассмотрены некоторые классы бесконечномерных банаховых пространств с интегрируемыми производными. Для нерефлексивных пространств получены лемма про компактность и основные топологические свойства данных пространств. Розглянуто деякі класи нескінченновимірних банахових просторів з інтегрованими похідними. Для нерефлексивних просторів одержано лемму про компактність та основні топологічні властивості даних просторів. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2017-09-08 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/109780 System research and information technologies; No. 1 (2008); 127-143 Системные исследования и информационные технологии; № 1 (2008); 127-143 Системні дослідження та інформаційні технології; № 1 (2008); 127-143 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/109780/104821 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Kasyanov, P. Mel’nik, V. Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1 |
| title | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1 |
| title_alt | On some topological properties for special classes of Banach spaces. Part 1 О некоторых топологических свойствах специальных классов банаховых пространств. Часть 1 |
| title_full | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1 |
| title_fullStr | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1 |
| title_full_unstemmed | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1 |
| title_short | Про деякі топологічні властивості спеціальних класів банахових просторів. Частина 1 |
| title_sort | про деякі топологічні властивості спеціальних класів банахових просторів. частина 1 |
| url | https://journal.iasa.kpi.ua/article/view/109780 |
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