On Principle of Equicontinuity
The main purpose of this paper is to prove some results of uniform boundedness principle type without the use of Baire’s category theorem in certain topological vector spaces; this provides an alternate route and important technique to establish certain basic results of functional analysis. As appli...
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2007
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| Cite this: | On Principle of Equicontinuity / Abdul Rahim Khan // Электронное моделирование. — 2007. — Т. 29, № 5. — С. 23-32. — Бібліогр.: 30 назв. — рос. |
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| citation_txt | On Principle of Equicontinuity / Abdul Rahim Khan // Электронное моделирование. — 2007. — Т. 29, № 5. — С. 23-32. — Бібліогр.: 30 назв. — рос. |
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| description | The main purpose of this paper is to prove some results of uniform boundedness principle type without the use of Baire’s category theorem in certain topological vector spaces; this provides an alternate route and important technique to establish certain basic results of functional analysis. As applications, among other results, versions of the Banach—Steinhaus theorem and the Nikodym boundedness theorem are obtained.
Обоснованы некоторые результаты типа принципа однородной ограниченности без использования теоремы категории Байера в некоторых топологических векторных пространствах, что обеспечивает альтернативный способ и важную методику для получения результатов функционального анализа. Получены также версии теоремы Банаха — Штейнхауза и теоремы ограниченности Никодима.
Обгрунтовано деякі результати типу принципу однорідної обмеженості без використання теореми категорії Байєра у деяких топологічних векторних просторах, що забезпечує альтернативний спосіб та важливу методику для отримання результатів функціонального аналізу. Отримано також версії теореми Банаха—Штейнхауза та теореми обмеженості Нікодима.
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Abdul Rahim Khan
Department of Mathematical Sciences
King Fahd University of Petroleum and Minerals
(Saudi Arabia, Dhahran 31261, e-mail: arahim@kfupm.edu.sa)
On Principle of Equicontinuity
(Recommended by Prof. E. Dshalalow)
The main purpose of this paper is to prove some results of uniform boundedness principle type
without the use of Baire’s category theorem in certain topological vector spaces; this provides an
alternate route and important technique to establish certain basic results of functional analysis. As
applications, among other results, versions of the Banach—Steinhaus theorem and the Nikodym
boundedness theorem are obtained.
Îáîñíîâàíû íåêîòîðûå ðåçóëüòàòû òèïà ïðèíöèïà îäíîðîäíîé îãðàíè÷åííîñòè áåç èñïîëü-
çîâàíèÿ òåîðåìû êàòåãîðèè Áàéåðà â íåêîòîðûõ òîïîëîãè÷åñêèõ âåêòîðíûõ ïðîñòðàíñòâàõ,
÷òî îáåñïå÷èâàåò àëüòåðíàòèâíûé ñïîñîá è âàæíóþ ìåòîäèêó äëÿ ïîëó÷åíèÿ ðåçóëüòàòîâ
ôóíêöèîíàëüíîãî àíàëèçà. Ïîëó÷åíû òàêæå âåðñèè òåîðåìû Áàíàõà — Øòåéíõàóçà è òåîðåìû
îãðàíè÷åííîñòè Íèêîäèìà.
K e y w o r d s: principle of equicontinuity, Banach—Steinhaus theorem, locally convex space,
thick set.
1. Introduction. The classical uniform boundedness principle asserts: if a se-
quence { f n } of continuous linear transformations from a Banach space X into a
normed space Y is pointwise bounded, then { f n } is uniformly bounded. The
proof of this result is most often based on the Baire’s category theorem (e. g. see
Theorem 4.7—3 [18] and Theorem 3.17 [26]); the interested reader is referred to
[10] for a new approach in this context. Several authors have sought proof of this
type of results without Baire’s theorem in various settings (see, for example, [4,
16], [23], [27]).
In 1933, Nikodym [21] proved: If a family M of bounded scalar measures
on a �-algebra A is setwise bounded, then the family M is uniformly bounded.
This result is a striking improvement of the uniform boundedness principle in the
space of countably additive measures on A; a Baire category proof of this theo-
rem may be found in [9, IV. 9.8, p. 309]. Nikodym theorem has received a great
deal of attention and has been generalized in several directions [5, 8, 19, 20, 28];
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 23
in particular, the proofs of this result without category argument for finitely ad-
ditive measures with values in a Banach space (quasi-normed group) are pro-
vided by Diestel and Uhl [6, Theorem 1, p. 14] and Drewnoski [7, Theorem 1],
respectively. For other related generalizations of this theorem, we refer to the
bibliography in [6].
Recently, Nygaard [22, 23] has used the notion of a «thick set» to prove the uni-
form bounded principle for transformations on a thick subset of a Banach space X
with values in another Banach space Y. The concept of a thick set goes back to the
ideas of Kadets and Fonf [12, 13, 15]. It is worth pointing out that the concept of
«thick sets» heavily depends on the dual of X and the development of their theory es-
sentially relies on the Hahn—Banach separation theorem in X. The broader class of
«thick sets» contains as a subclass the class of second category sets.
In this paper, certain aspects of the development of the uniform bounded-
ness principle are discussed; in particular, results of the type of uniform boun-
dedness principle are proved on a domain of second category and beyond with-
out employing Baire’s category argument. First, we prove a general principle of
equicontinuity for maps on a topological vector space of the second category
with values in another topological vector space. A similar result is obtained for
transformations on «thick sets» of a complete locally convex space X satisfying
the property (N) and taking values in a locally convex space Y ; this generalizes
the uniform boundedness principle of Nygaard [23] to a class of locally convex
spaces. An analogue of the new result is given for maps from X *
into Y *
. Some
versions of the Banach—Steinhaus theorem and the Nikodym boundedness the-
orem are also given.
2. Notations and preliminaries. Let P be a family of seminorms on a
Hausdorff locally convex space X. Let B x X p xX � � �{ : ( ) 1for each p P� } and
S x X p xX � � �{ : ( ) 1 for each p P� }; see [3, part III, p. 13, 14)]. The strong dual
X *
of X is a locally convex space (details may be found in [3, part IV, p. 14—23].
For our purposes, it would be enough to consider the following: suppose that �
is a family of bounded subsets of X. The pair (�, |·|) induces a locally convex to-
pology on X *
via the family P*
of seminorms
p x x x x A* * *
( ) sup{ ( ) :� � , A��}.
Similarly, if Q is a family of seminorms on a locally convex space Y, then Q*
will
be the induced family of seminorms defining the locally convex topology on Y *
.
Let X and X *
be in duality. The polar of A X� and B X�
*
are, respec-
tively, defined by
A x X x x
x A
0
1� � �
�
�
�
�
* * *
: sup ( ) ,
Abdul Rahim Khan
24 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
B x X x x
x B
0
1� � �
�
{ : sup ( ) }
*
*
,
where we consider X to be embedded in X **
, bidual of X [30].
Locally convex spaces provide a very general framework for the Hahn—
Banach theorem and its consequences; in particular, we shall need the following
separation result.
Proposition 2.1 [27, Prop. 13, p. 173]. Let A be a closed and absolutely con-
vex subset of a Hausdorff locally convex space X and x A� . Then there exists
x X* *
� such that x x x y y A* *
( ) sup{ ( ) : }� � �1 .
In what follows we will use the terminology of Nygaard [22, 23].
A subset A of a normed space X is norming for X *
if for some � > 0,
inf sup ( )
*
*
*
x S x AX
x x
�
�
� �. Analogously, a subset B of X *
is norming for X (or
�
*
-norming) if for some � > 0, inf sup ( )
*
*
x S x BX
x x
�
�
� �. We say a subset A of X is
thin if it is countable union of an increasing sequence of sets which are
non-norming for X *
. A set which is not thin, is called a thick set.
The concept of�
*
-thin and�
*
-thick sets can be defined in the same way.
A set A in a complex vector space X is norming if for some � > 0,
co rA B
r
X
�
�
�
�
�
�
�
�
�
�
1
� � . However, we shall employ co A B X( )� �� for simplicity.
It will be interesting to formulate the above definitions in the context of an
arbitrary locally convex space.
Let G be a commutative group. A non-negative valued function q on G is
said to be a quasi-norm if it has the following properties for any x, y in G:
(i) q (0) = 0;
(ii) q (x) = q (– x);
(iii) q (x + y) � q (x) + q (y).
The relationship of functional analysis and measure theory is not so easy to
understand (for some connections, we refer to [14]). Recently, Abrahamsen et al.
[1] have established in Prop. 3.2, boundedness of a vector measure by utilizing
the concept of a thick set; thereby reflecting growing interaction between these
two subjects. Consequently, such an interplay will play a part here.
Let G be a commutative Hausdorff topological group andR a ring of subsets
of a set X. A function µ :R�G is said to be: (i) measure if µ (�) = 0 and µ (E F� ) =
= µ (E) + µ (F) where E and F are in R with E� F = � (ii) exhaustive if for every
sequence {En} of pairwise disjoint sets in R, lim ( )
n
nE
�
�! 0.
On principle of equicontinuity
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 25
The notion of a submeasure has been extensively studied by Drewnoski [7, 8
and the references therein]. The applications of this concept are enormous [8,
24]. Group-valued submeasures have been introduced by Khan and Rowlands
[17] and their work has been further investigated by Avallone and Valente [2].
Let G be a commutative lattice group (�-group). A quasi-norm q on G is an
�-quasi-norm if q (x) � q (y) for all x, y in G with |x| � |y| where |x| = x+
+ x–
. An
�-quasi-norm generates a locally solid group topology on G [14, Prop. 2 2C].
Following Khan and Rowlands [17], a G-valued function µ onR is a submeasure
if! �( ) �0, ! ! !( ) ( ) ( )E F E F� � " for all E, F in R with E F� �� and! ( )E �
�! ( )F for all E, F in R with E � F. Clearly, in this case µ (E) � 0 for all E in R.
3. Main results. Khan and Rowlands [16] have obtained the following im-
provement of Theorem 2 due to Dane
�
s [4].
Theorem A. [16, Corollary 1]. Let X be a topological vector space, {xn} a
sequence in X such that lim
n
nx
�
�0 , and {pn} a sequence of real sub-additive
functionals on X satisfying the condition: «there exists a sequence {ak} of real
numbers, ak � " as k� , such that, for each k, n = 1, 2, ..., the set Bk,n = {x�
� X : pn(x) � ak} is closed in X».
If lim sup(sup ( ))
n x U
np x
�
� " for each neighbourhood U of 0 in X, then the
set Z z X p x z p x z
n
n n
n
n n� � " � " # � " { : lim sup ( ) lim sup ( ) }or is a resid-
ual G
�
-set in X.
The following example reveals that Theorem A is not true, in general, if Z is
replaced by either Z
+
or Z
–
where Z
+
= {x� X: x > 0}, Z
–
= {x� X : x < 0} and
Z = Z
+
� Z
–
.
Let X be the usual space of real numbers. We assume that xn = 0 for each
n�N. Define p x n xn ( ) � (x X� , n�N). Here Z
#
�� and so Z# can not be re-
sidual G
�
-set while Z
+
is a residual G
�
-set, for X \ Z
+
= {0} is of first category in
X. Thus, either Z
+
or Z
#
can be a residual G
�
-set.
As an application of Theorem A, we establish a principle of equicontinuity
in the following result; this leads to an alternative proof of the Banach—
Steinhaus theorem given by Rudin [25].
Theorem 1. (Principle of equicontinuity). Let X be a topological vector
space of the second category, Y a Hausdorff topological vector space and {fn} a
sequence of continuous linear transformations of X into Y such that the set
{fn(x)} is bounded for each x � X. Then the sequence {fn} is equicontinuous.
P r o o f. Let the topology of Y be determined by a family {q i Ii : � } of
F-seminorms (definition and details may be found in [29, p. 1—3]). Suppose that
the sequence {fn} is not equicontinuous. Then for some continuous quasi-norm
Abdul Rahim Khan
26 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
qi
0
, which for the sake of simplicity we denote by q, and any $-neighbourhood U
of 0 in X, there exist a sequence {xn} in U and a sequence of integers
n n nk k k
1 2 3
% % % ··· such that q f xn nk
( ( )) > k (k = 1, 2, ...). It follows that
lim sup(sup ( ( )))
n x U
nq f x
�
� " . The functionals q f n0
(n = 1, 2, ...) satisfy the con-
ditions of Theorem A (taking xn = 0 for all n = 1, 2, ...), and so the set
Z z X q f z
n
n� � �" { : lim sup ( ( )) }
is a residual G
�
-set in X. Thus X \ Z is of the first category. Since X is of the
second category, it follows that Z is non-empty; this implies that there is a point
z0� X such that lim sup ( ( ))
n
nq f z
0
� " . This contradicts the hypothesis. Thus
the sequence {fn}is equicontinuous.
An immediate consequence of the above theorem is given below.
Theorem 2. (Banach—Steinhaus theorem). Let X and Y be as in Theorem 1
and let {fn} be a sequence of continuous linear transformations of X into Y such
that f x f x
n
n( ) lim ( )�
�
exists for each x� X. Then f is a continuous linear trans-
formation of X into Y .
P r o o f. Clearly, f is a homomorphism and the sequence {fn (x)} is bounded.
By Theorem 1, the sequence { fn} is equicontinuous. Let V be any neighbour-
hood of 0 in Y. Then there exist a closed neighbourhood V0& V and a neighbour-
hood U of 0 in X such that fn(U) & V0 (n = 1, 2, ...). Now, for any x�U,
f x f x V V
n
n( ) lim ( )� � �
�
0 0
and so f (U) & V ; that is, f is continuous.
For X, as in Theorem 1, let M � X *
be�
*
-bounded (i.e., sup{|f (x)| : f � M} <
% for every x � X). Then M is pointwise bounded in X *
and so bounded by
Theorem 1.
In the same way, some other results purely dependent on the classical uniform
boundedness principle can be adopted from [11, 26, 30] in this general setting.
As another application of Theorem A, we indicate how the Banach—
Steinhaus theorem on condensation of singularities [27, Corollary 3, p. 121] may
be derived from it.
Theorem 3. Let {Un,m : n, m = 1, 2, ...} be a double sequence of bounded lin-
ear transformations of a Banach space X into a Banach space Y such that for each
m = 1, 2, ..., lim sup
,n n mU � " . Then there is a set S of the second category in X
such that, for each x in S and each m = 1, 2, ..., lim sup ( )
,
n
n mU x � " .
On principle of equicontinuity
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 27
P r o o f. For each positive integer m, n define p x U xn m n m, ,
( ) ( )� ( )x X� . It
is easy to see that each pn m,
is a continuous sub-additive functional on X. For
each positive integer m, define
Z z X p zm
n
n m� � �" { : lim sup ( ) }
,
and
Z Zm
m
�
�
1
� .
The condition lim sup
,
n
n mU � " implies that, for each m = 1, 2, ...,
lim sup( sup ( ))
,
n x U
n mp x
�
� " for each neighbourhood U of 0 in X, and therefore
by Theorem A (with xn = 0 for all positive integers n), Zm is a residual G
�
-set. It
follows that Z is a residual G
�
-set. Since X is a Banach space, therefore
Z z X U z
n
n m� � �" { :lim sup ( )
,
for m = 1, 2, ...} is of second category and is
the desired set S.
A locally convex space in which a norm is available, is said to have the prop-
erty (N). For example, a normed space and the space (X *
,�
*
) where X is a lo-
cally convex space have the property (N).
In the remainder of this section it is assumed that X is a complete locally
convex space with the property (N).
We need the following pair of lemmas.
L e m m a 1. The following statements are equivalent for a subset A of X:
a) A is norming for X *
;
b) co (±A) is norming for X *
;
c) there exits a � > 0 such that co (±A) �'�BX.
P r o o f. The only non-trivial implication is (a)( (c).
Assume that co (± A) � �BX for all � > 0. Consider a sequence {xn} in
X \ co (A) converging to 0. For each n, xn� co (± A), an absolutely convex subset
of X, so by Prop. 2.1 (see also Theorem 4.25 in [11]) there exists x Xn
* *
� such
that
x x x a x an n
a co A
n
a A
n
*
( )
* *
( ) sup ( ) sup ( )� �
� � �
.
Now using (a), we may obtain � > 0 satisfying
x x x an
x S a A
n
n X
* *
( ) inf sup ( )
*
*
� �
� �
� .
Plainly the choice of {xn} implies that x xn n
*
( ) % � for all � > 0 and n � n0. This
contradiction proves the result.
Abdul Rahim Khan
28 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
The following analogous result for the dual space X *
is easy to verify.
L e m m a 2. The following statements are equivalent for a subset B of X *
:
a) B is norming for X;
b) co (± B) is norming for X;
c) there exists � > 0 such that co
�
*
(± B) �'� B
X * .
L e m m a 3. If A is a subset of the second category in X, then A is thick.
P r o o f. Let {Ai} be an increasing sequence with A Ai
i
�
�
1
� . As A is of second
category, some Am contains a ball S xr ( ). Hence, it follows that S co Ar m( ) ( )0 & � .
This implies, by Lemma 1 (with � = 1), Am is norming. Since {Ai} is arbitrary,
therefore A must be thick.
The classical uniform boundedness principle holds beyond sets of the sec-
ond category; this is the case with the set S of characteristic functions in the unit
sphere of the function space B ( )A where A is a �-algebra of sets [6]. Note that S
is merely nowhere dense. We continue this theme and generalize Theorems 1
and 2 and Prop. 2.2 of Nygaard [23] in the sense that the domain of transforma-
tions is a thick set in X and its dual space X *
. Our methods are based on those
used by Nygaard [22, 23].
Theorem 4. Let A be a thick subset of X. Suppose that Y is a Hausdorff lo-
cally convex space and { fn} a sequence of continuous linear transformations of
X into Y such that { fn (x)} is bounded for each x � A. Then the sequence { fn} is
equicontinuous.
P r o o f. Suppose that {fn} is pointwise bounded on A, that is,
sup ( ( ))
n
np f x % for all x� A and each p � P. Put A x A p f x mm n n� � �{ :sup ( ( ))
for each p� P}. The sequence {Am} of sets is increasing with A Ai
i
�
�
1
� . As A is
thick, some Ak is norming. Thus, by Lemma 1, there exists a � > 0 such that
�B co AX k& �( ). This together with the definition of Am implies that �p f n( ) �
�
�
sup ( ( ))
x S
n
X
p f x
�
� �
� �
sup ( ( ))
( )x co A
n
k
p f x k. Hence,sup ( )
n
np f
k
� %
�
as desired.
R e m a r k. Theorem 4 extends Prop. 2.2 of Nygaard [23].
Theorem 5. Let B be a thick subset of X *
. Suppose that Y is a Hausdorff lo-
cally convex space and { f n
*
} a sequence of continuous linear transformations of
X *
into Y *
such that { ( )}
* *f xn is bounded for each x*
in B. Then the sequence
{ }
*f n is equicontinuous.
On principle of equicontinuity
ISSN 0204–3572. Ýëåêòðîí. ìîäåëèðîâàíèå. 2007. Ò. 29. ¹ 5 29
P r o o f. Follows pattern of the proof of Theorem 4; the only difference is
that we consider
A x B q f x mm
n
n� � �{ :sup ( ( ))
* * * *
for each q Q* *
}�
and use Lemma 2 and the�
*
-continuity of f n
*
.
The proofs of the following corollaries follow pattern of the proof of Theo-
rem 2 and so will be omitted.
Corollary 1. Let X, A and Y be as in Theorem 4 and {fn} be a sequence of
continuous linear transformations of X into Y such that f x f x
n
n( ) lim ( )�
�
for
each x � X. Then f is a continuous linear transformation of X into Y .
Corollary 2. Let X *
, B and Y *
be as in Theorem 5 and { f n
*
} be a se-
quence of continuous linear transformations of X *
into Y *
such that f x* *
( ) �
�
�
lim ( )
* *
n
nf x exists for each x*
� X *
. Then f *
is a continuous linear transforma-
tion of X *
into Y *
.
We now establish the Nikodym boundedness theorem in more general set-
tings in relation to the domain, range and nature of mappings.
Theorem 1 due to Drewnoski [7] is proved in the context of a quasi-normed
group; we observe that his proof can be readily modified to the case of any com-
mutative Hausdorff topological group G to obtain a principle of equicontinuity
type result for group measures as follows.
Theorem 6. Let M be a family of exhaustive G-valued measures on a �-ring
R such that for each E�R,{ ( ) : }! !E M� is a bounded subset of G. Then {µ (E) :
E�R, µ � M} is a bounded subset of G.
The assumption that R is a �-ring is essential in the above theorem [7, Ex-
ample, p. 117].
Valuable contributions have been made in special but very important field
of submeasures with values in a commutative �-group [2, 17 and the references
therein]. In the next result, we prove Theorem 6 for group-valued submeasures
to get the following principle of equicontinuity which generalizes Theorem 1 of
Drewnowski [7].
Theorem 7. Let (G, q) be an �-quasi-normed group and M be a family of
G-valued submeasures on a �-ring R such that
sup ( ( ))
!
!
�
% "
M
q E
for each E in R. Then sup ( ( ))
!
!
�
�
% "
M
E
q E
R
.
Abdul Rahim Khan
30 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
P r o o f. Let H be the group of all G-valued mappings on M. Clearly, H is a
commutative partially ordered group, the order being f � g if and only if f (µ) �
� g (µ) for all µ � M. Define the functional � on H by
� !
!
( ) sup ( ( ))f q f
M
�
�
.
Note that � is an extended real-valued quasi-norm on H with � �( ) ( )f g�
for 0� f � g. Define a mapping ) : R� H by
) ! !( ) ( ) ( )E E� .
Clearly, ) is an H-valued submeasure on R.
Suppose not; then with the above notation, sup ( ( ))
E
E
�
� "
R
� ) . Thus, for each
positive integer n, there exists a set En inR such that� )( ( ))E nn � . Let E En
n
�
�
1
� .
Now E�R and � )( ( ))E � " . This implies that sup ( ( )) ,
!
!q E � " which con-
tradicts the hypothesis. Hence, sup ( ( ))
!
!
�
�
M
E
q E
R
is finite.
Finally, every �-algebra of sets on a finite set S is a topology but not con-
versely. Thus, the result to follow extends the domain of maps in [6, Corollary 2,
p. 16] and [23, Prop. 2.1], simultaneously.
Theorem 8. Let A be a thick subset of X. If {fn} is a sequence of continuous
linear functionals on X such that {fn(x)} is bounded for each x in A, then the se-
quence {fn} is equicontinuous.
P r o o f. Take Y as the space of scalars in the proof of Theorem 4.
Îáãðóíòîâàíî äåÿê³ ðåçóëüòàòè òèïó ïðèíöèïó îäíîð³äíî¿ îáìåæåíîñò³ áåç âèêîðèñòàííÿ
òåîðåìè êàòåãî𳿠Áàéºðà ó äåÿêèõ òîïîëîã³÷íèõ âåêòîðíèõ ïðîñòîðàõ, ùî çàáåçïå÷óº àëüòåð-
íàòèâíèé ñïîñ³á òà âàæëèâó ìåòîäèêó äëÿ îòðèìàííÿ ðåçóëüòàò³â ôóíêö³îíàëüíîãî àíàë³çó.
Îòðèìàíî òàêîæ âåðñ³¿ òåîðåìè Áàíàõà—Øòåéíõàóçà òà òåîðåìè îáìåæåíîñò³ ͳêîäèìà.
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Ïîñòóïèëà 25.09.06
Abdul Rahim Khan
32 ISSN 0204–3572. Electronic Modeling. 2007. V. 29. ¹ 5
|
| id | nasplib_isofts_kiev_ua-123456789-101813 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0204-3572 |
| language | English |
| last_indexed | 2025-12-07T17:11:39Z |
| publishDate | 2007 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Abdul Rahim Khan 2016-06-07T14:29:41Z 2016-06-07T14:29:41Z 2007 On Principle of Equicontinuity / Abdul Rahim Khan // Электронное моделирование. — 2007. — Т. 29, № 5. — С. 23-32. — Бібліогр.: 30 назв. — рос. 0204-3572 https://nasplib.isofts.kiev.ua/handle/123456789/101813 The main purpose of this paper is to prove some results of uniform boundedness principle type without the use of Baire’s category theorem in certain topological vector spaces; this provides an alternate route and important technique to establish certain basic results of functional analysis. As applications, among other results, versions of the Banach—Steinhaus theorem and the Nikodym boundedness theorem are obtained. Обоснованы некоторые результаты типа принципа однородной ограниченности без использования теоремы категории Байера в некоторых топологических векторных пространствах, что обеспечивает альтернативный способ и важную методику для получения результатов функционального анализа. Получены также версии теоремы Банаха — Штейнхауза и теоремы ограниченности Никодима. Обгрунтовано деякі результати типу принципу однорідної обмеженості без використання теореми категорії Байєра у деяких топологічних векторних просторах, що забезпечує альтернативний спосіб та важливу методику для отримання результатів функціонального аналізу. Отримано також версії теореми Банаха—Штейнхауза та теореми обмеженості Нікодима. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Электронное моделирование Математические методы и модели On Principle of Equicontinuity Article published earlier |
| spellingShingle | On Principle of Equicontinuity Abdul Rahim Khan Математические методы и модели |
| title | On Principle of Equicontinuity |
| title_full | On Principle of Equicontinuity |
| title_fullStr | On Principle of Equicontinuity |
| title_full_unstemmed | On Principle of Equicontinuity |
| title_short | On Principle of Equicontinuity |
| title_sort | on principle of equicontinuity |
| topic | Математические методы и модели |
| topic_facet | Математические методы и модели |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/101813 |
| work_keys_str_mv | AT abdulrahimkhan onprincipleofequicontinuity |