Some applications of the open mapping theorem in locally convex cones
UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones.  Specially, we prove some relations between an open linear operator and its adjoint in  $uc$-cones (locally convex cones which their c...
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Institute of Mathematics, NAS of Ukraine
2021
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| author | Jafarizad, S. Ranjbari, A. Jafarizad, S. Ranjbari, A. |
| author_facet | Jafarizad, S. Ranjbari, A. Jafarizad, S. Ranjbari, A. |
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| description | UDC 515.12
We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones.  Specially, we prove some relations between an open linear operator and its adjoint in  $uc$-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).
  |
| doi_str_mv | 10.37863/umzh.v73i3.222 |
| first_indexed | 2026-03-24T02:02:19Z |
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| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
DOI: 10.37863/umzh.v73i3.222
UDC 515.12
S. Jafarizad, A. Ranjbari (Univ. Tabriz, Iran)
SOME APPLICATIONS OF THE OPEN MAPPING THEOREM
IN LOCALLY CONVEX CONES
ДЕЯКI ЗАСТОСУВАННЯ ТЕОРЕМИ ПРО ВIДКРИТЕ ВIДОБРАЖЕННЯ
У ЛОКАЛЬНО-ОПУКЛИХ КОНУСАХ
We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones.
Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which
their convex quasi-uniform structures are generated by one element).
Показано, що неперервний вiдкритий лiнiйний оператор зберiгає повноту та бочкуватiсть у локально-опуклих
конусах. Зокрема, доведено деякi спiввiдношення мiж вiдкритим лiнiйним оператором та його сумiжним у uc-
конусах (локально-опуклих конусах, у яких опуклi квазiрiвномiрнi структури генеруються одним елементом).
1. Introduction. The theory of locally convex cones deals with ordered cones that are not necessarily
embeddable in vector spaces. A topological structure is introduced using an order theoretical concept
or a convex quasiuniform structure. In this paper we use the latter. These cones developed in [4, 8].
For recent researches see [1, 9]. We shall review some of the concepts and refer to [4, 8] for details.
A cone is defined to be a commutative monoid \scrP together with a scalar multiplication by non-
negative real numbers satisfying the same axioms as for vector spaces; that is, \scrP is endowed with
an addition (a, b) \mapsto \rightarrow a + b : \scrP \times \scrP \rightarrow \scrP which is associative, commutative and admits a neutral
element 0 \in \scrP , and with a scalar multiplication (r, a) \mapsto \rightarrow r \cdot a : \BbbR + \times \scrP \rightarrow \scrP satisfying the usual
associative and distributive properties, where \BbbR + is the set of nonnegative real numbers. We have
1 \cdot a = a and 0 \cdot a = 0 for all a \in \scrP .
Let \scrP be a cone. A convex quasiuniform structure on \scrP is a collection U of convex subsets
U \subseteq \scrP 2 = \scrP \times \scrP such that the following properties hold:
(U1) \Delta \subseteq U for every U \in U, where \Delta =
\bigl\{
(a, a) : a \in \scrP
\bigr\}
;
(U2) for all U, V \in U there is a W \in U such that W \subseteq U \cap V ;
(U3) \lambda U \circ \mu U \subseteq (\lambda + \mu )U for all U \in U and \lambda , \mu > 0;
(U4) \alpha U \in U for all U \in U and \alpha > 0.
Here, for U, V \subseteq \scrP 2, by U \circ V we mean the set of all (a, b) \in \scrP 2 such that there is some c \in \scrP
with (a, c) \in U and (c, b) \in V.
Let \scrP be a cone and U be a convex quasiuniform structure on \scrP . We shall say (\scrP ,U) is a locally
convex cone if
(U5) for each a \in \scrP and U \in U there is some \lambda > 0 such that (0, a) \in \lambda U.
c\bigcirc S. JAFARIZAD, A. RANJBARI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3 425
426 S. JAFARIZAD, A. RANJBARI
With every convex quasiuniform structure U on \scrP we associate two topologies on \scrP : the
neighborhood bases for an element a in the upper and lower topologies are given by the sets
U(a) = \{ b \in \scrP : (b, a) \in U\} , resp., (a)U = \{ b \in \scrP : (a, b) \in U\} , U \in U.
The common refinement of the upper and lower topologies is called symmetric topology. A neigh-
borhood base for a \in \scrP in this topology is given by the sets
U(a)U = U(a) \cap (a)U, U \in U.
The extended real numbers system \BbbR = \BbbR \cup \{ +\infty \} is a cone endowed with the usual algebraic
operations, in particular a + (+\infty ) = +\infty for all a \in \BbbR , \alpha \cdot (+\infty ) = +\infty for all \alpha > 0 and
0 \cdot (+\infty ) = 0. We set \widetilde \scrV = \{ \~\varepsilon : \varepsilon > 0\} , where
\~\varepsilon =
\bigl\{
(a, b) \in \BbbR 2
: a \leq b+ \varepsilon
\bigr\}
.
Then \widetilde \scrV is a convex quasiuniform structure on \BbbR and (\BbbR , \widetilde \scrV ) is a locally convex cone. For a \in \BbbR
the intervals ( - \infty , a+ \varepsilon ] are the upper and the intervals [a - \varepsilon ,+\infty ] are the lower neighborhoods,
while for a = +\infty the entire cone \BbbR is the only upper neighborhood, and \{ +\infty \} is open in the
lower topology. The symmetric topology is the usual topology on \BbbR with as an isolated point +\infty .
For cones \scrP and \scrQ , a mapping T : \scrP \rightarrow \scrQ is called a linear operator if T (a + b) = T (a) +
+ T (b) and T (\alpha a) = \alpha T (a) hold for all a, b \in \scrP and \alpha \geq 0. If both (\scrP ,U) and (\scrQ ,\scrW ) are
locally convex cones, the operator T is called (uniformly) continuous if for every W \in \scrW one can
find U \in U such that (T \times T )(U) \subseteq W. Uniform continuity implies continuity for the operator
T : \scrP \rightarrow \scrQ with respect to the upper, lower and symmetric topologies on \scrP and \scrQ , respectively.
A linear functional on \scrP is a linear operator \mu : \scrP \rightarrow \BbbR . We note that \mu : \scrP \rightarrow \BbbR is continuous
if and only if there is U \in U such that \mu (a) \leq \mu (b) + 1 for all (a, b) \in U. We denote the set of all
linear functional on \scrP by \scrL (\scrP ) (the algebraic dual of \scrP ). For a subset F of \scrP 2, we define polar
F \circ as follows:
F \circ =
\bigl\{
\mu \in \scrL (\scrP ) : \mu (a) \leq \mu (b) + 1 for all (a, b) \in F
\bigr\}
.
The dual cone \scrP \ast of a locally convex cone (\scrP ,U) consists of all continuous linear functionals
on \scrP and is the union of all polars U\circ of neighborhoods U \in U.
2. Some applications of the open mapping theorem. An open linear operator was defined
in [2] as follows:
Definition 2.1. Let (\scrP ,U) and (\scrQ ,\scrW ) be locally convex cones. A linear operator T : (\scrP ,U) \rightarrow
\rightarrow (\scrQ ,\scrW ) is called (uniformly) open if for every U \in U one can find W \in \scrW such that W \subseteq
\subseteq (T \times T )(U).
If T : (\scrP ,U) \rightarrow (\scrQ ,\scrW ) is open, then it is open under the upper, lower and symmetric topologies.
Also if T : \scrP \rightarrow \scrQ is open, then T is surjective (see [2]).
A Cauchy net in locally convex cones was defined in [5] as follows.
Definition 2.2. Let (\scrP ,U) be a locally convex cone. A net (x\alpha )\alpha \in \scrI in \scrP is called lower (upper)
Cauchy if for every U \in U there is some \gamma U \in \scrI such that (x\beta , x\alpha ) \in U (resp., (x\alpha , x\beta ) \in U) for
all \alpha , \beta \in \scrI with \beta \geq \alpha \geq \gamma U . Also (x\alpha )\alpha \in \scrI is called symmetric Cauchy if for each U \in U there
is some \gamma U \in \scrI such that (x\beta , x\alpha ) \in U for all \alpha , \beta \in \scrI with \alpha , \beta \geq \gamma U .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
SOME APPLICATIONS OF THE OPEN MAPPING THEOREM IN LOCALLY CONVEX CONES 427
We call that a net (xi)i\in \scrI in (\scrP ,U) is lower (upper) convergent to x \in \scrP if for every U \in U
there is some \gamma U \in \scrI such that (x, xi) \in U (resp., (xi, x) \in U) for all i \geq \gamma U . Also (xi)i\in \scrI is
called symmetric convergent to x if for each U \in U there is some \gamma U \in \scrI such that (xi, x) \in U and
(x, xi) \in U for all i \geq \gamma U .
A locally convex cone (\scrP ,U) is called lower (upper or symmetric) complete if every lower (resp.,
upper or symmetric) Cauchy net in \scrP converges in the lower (resp., upper or symmetric) topology
to some element of \scrP .
Proposition 2.1. Let (\scrP ,U) and (\scrQ ,\scrW ) be locally convex cones. If there is a one-to-one open
continuous linear mapping of \scrP into \scrQ , and \scrP is lower (upper or symmetric) complete, then so
is \scrQ .
Proof. We prove for the symmetric case. Let the mapping be T and let (yi)i\in \scrI be a symmetric
Cauchy net in \scrQ . Since every open linear mapping is surjective (see [2]), for every i \in \scrI , there
exists xi \in \scrP such that yi = T (xi). We show that (xi)i\in \scrI is a Cauchy net in \scrP . Let U \in U be
arbitrary. By the openness of T, there exists W \in \scrW such that W \subseteq (T \times T )(U). Since (yi)i\in \scrI is
symmetric Cauchy, there is \gamma W such that (yi, yj) \in W for all i, j \geq \gamma W . Hence
(yi, yj) =
\bigl(
T (xi), T (xj)
\bigr)
\in (T \times T )(U)
for all i, j \geq \gamma W . Since T is one-to-one, (xi, xj) \in U for all i, j \geq \gamma W , i.e., (xi)i\in \scrI is symmetric
Cauchy. Since \scrP is symmetric complete, there is x \in \scrP such that (xi)i\in \scrI converges to x in the
symmetric topology. The continuity of T renders that (T (xi))i\in \scrI is convergent to T (x) \in \scrQ . This
shows that (yi)i\in \scrI is convergent to T (x) in the symmetric topology.
The notions of a barrel and a barreled locally convex cone were introduced in [10] as follows: a
barrel is a convex subset B of \scrP 2 with the following properties:
(B1) For every b \in \scrP there is U \in U such that for every a \in U(b)U there is \lambda > 0 such that
(a, b) \in \lambda B.
(B2) For all a, b such that (a, b) /\in B there is \mu \in \scrP \ast such that \mu (c) \leq \mu (d) + 1 for all
(c, d) \in B and \mu (a) > \mu (b) + 1.
A locally convex cone (\scrP ,U) is said to be barreled if for every barrel B \subseteq \scrP 2 and every element
b \in \scrP there are neighborhood U \in U and \lambda > 0 such that (a, b) \in \lambda B for all a \in U(b)U.
Theorem 2.1. Let (\scrP ,U) and (\scrQ ,\scrW ) be two locally convex cones. Let T be a linear continuous
and open mapping of \scrP into \scrQ . If (\scrP ,U) is barrelled, then (\scrQ ,\scrW ) is barrelled too.
Proof. Let B be a barrel in \scrQ 2 and y \in \scrQ . We show that there are W \in \scrW and \lambda > 0 such
that (x, y) \in \lambda B for all x \in W (y)W. There is an element b \in \scrP such that T (b) = y, because T is
surjective (see [2]). Since T is continuous, (T \times T ) - 1(B) is a barrel in \scrP 2 (see [6]). Since \scrP is
barrelled, there are U \in U and \lambda > 0 such that (a, b) \in \lambda (T \times T ) - 1(B) for all a \in U(b)U. There
is W \in \scrW such that
W \subseteq (T \times T )(U), (2.1)
because T is open by the hypothesis. Now let x \in W (y)W. We have, by (2.1),
W (y)W = W (T (b))W \subseteq T (U(b)U).
Hence there is a\prime \in U(b)U such that T (a\prime ) = x. Therefore (a\prime , b) \in \lambda (T \times T ) - 1(B) and so
(x, y) \in \lambda B.
Theorem 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
428 S. JAFARIZAD, A. RANJBARI
A locally convex cone (\scrP ,U) is called upper-barreled if for every barrel B \subseteq \scrP 2, there is
U \in U such that U \subseteq B (see [6]). Proposition 2.11 from [2] yields that, every open continuous
linear operator preserves the upper-barreledness between locally convex cones.
Let (\scrP ,U) and (\scrQ ,\scrW ) be locally convex cones and T : \scrP \rightarrow \scrQ be a linear operator. The adjoint
operator T \ast : \scrQ \ast \rightarrow \scrL (\scrP ) is defined as follows: for any \mu \in \scrQ \ast define the linear functional T \ast (\mu )
on \scrP by T \ast (\mu )(a) = \mu (T (a)) for all a \in \scrP . If T is continuous, then T \ast is a linear operator
Lemma 2.1. Let (\scrP ,U) and (\scrQ ,\scrW ) be locally convex cones and T : \scrP \rightarrow \scrQ be a linear
operator.
(1) If T is surjective, then T \ast is one-to-one.
(2) If \scrL (\scrP ) separates the elements of \scrP and T \ast is surjective, then T is one-to-one.
Proof. (1) Suppose T \ast (\mu 1) = T \ast (\mu 2). Then T \ast (\mu 1)(a) = T \ast (\mu 2)(a) for all a \in \scrP , i.e.,
\mu 1(T (a)) = \mu 2(T (a)) for all a \in \scrP . Since T is surjective, \mu 1(q) = \mu 2(q) for all q \in \scrQ . Hence
\mu 1 = \mu 2.
(2) Suppose T (x) = T (y). Let \mu \in \scrL (\scrP ). There is \mu 1 \in \scrQ \ast such that T \ast (\mu 1) = \mu . We have
\mu 1(T (x)) = \mu 1(T (y)) and so T \ast (\mu 1)(x) = T \ast (\mu 1)(y). Therefore \mu (x) = \mu (y) for all \mu \in \scrL (\scrP ).
Hence x = y, since \scrL (\scrP ) separates the elements of \scrP by the hypothesis.
Lemma 2.1 is proved.
Lemma 2.2. Let (\scrP ,U) and (\scrQ ,\scrW ) be locally convex cones and T from \scrP onto \scrQ be a linear
operator. Then for the adjoint operator T \ast we have
\widetilde T \ast (W \circ ) = (T \ast \times T \ast )(\widetilde W \circ ),
where \widetilde W \circ = \{ (\mu , \nu ) \in \scrQ \ast \times \scrQ \ast : \nu \in \mu +W \circ \} .
Proof. Let (\mu , \nu ) \in (T \ast \times T \ast )(\widetilde W \circ ). Then there is (\mu \prime , \nu \prime ) \in \widetilde W \circ such that \mu = T \ast (\mu \prime ) and
\nu = T \ast (\nu \prime ). Hence there is \Lambda \in W \circ such that \nu \prime = \mu \prime + \Lambda . Then T \ast (\nu \prime ) = T \ast (\mu \prime ) + T \ast (\Lambda ). So
\nu = \mu + T \ast (\Lambda ), i.e., (\mu , \nu ) \in \widetilde T \ast (W \circ ). Conversely, let (\mu , \nu ) \in \widetilde T \ast (W \circ ). Then \nu \in \mu + T \ast (W \circ ).
Hence there is \Lambda \in W \circ such that \nu = \mu +T \ast (\Lambda ). Thus T \ast - 1(\nu ) = T \ast - 1(\mu )+\Lambda (by Lemma 2.1 (1),
T \ast is invertible). Hence (T \ast - 1(\mu ), T \ast - 1(\nu )) \in \widetilde W \circ , i.e., (\mu , \nu ) \in (T \ast \times T \ast )(\widetilde W \circ ).
Lemma 2.2 is proved.
We recall the following results.
Lemma 2.3 ([2], Lemma 2.3). Let (\scrP ,U) and (\scrQ ,\scrW ) be locally convex cones and T : \scrP \rightarrow \scrQ
a linear operator. Then:
(1) for each subset F of \scrP 2, T \ast - 1(F \circ ) =
\bigl(
(T \times T )(F )
\bigr) \circ
,
(2) if the polar E\circ being taken in \scrQ \ast , for each subset E of \scrQ 2, T \ast (E\circ ) \subseteq
\bigl(
(T \times T ) - 1(E)
\bigr) \circ
and if T is invertible, then we have the inverse inclusion, i.e., T \ast (E\circ ) =
\bigl(
(T \times T ) - 1(E)
\bigr) \circ
.
Theorem 2.2 (Extension theorem [4], II.2.9). Let \scrQ be a subcone of a locally convex cone
(\scrP ,U). Then every continuous linear functional on \scrQ can be extended to a continuous linear func-
tional on \scrP .
A locally convex cone (\scrP ,U) is called a uc-cone whenever U = \{ \alpha U : \alpha > 0\} for some
U \in U. The uc-cones in locally convex cones play the role of normed spaces in topological vector
spaces. If (\scrP ,U) is a uc-cone and U = \{ \alpha U : \alpha > 0\} , then (\scrP \ast ,U\beta (\scrP \ast ,\scrP )) is a uc-cone, where
U\beta (\scrP \ast ,\scrP ) = \{ \alpha \widetilde U\circ : \alpha > 0\} (see [1]). If (\scrP ,U) and (\scrQ ,\scrW ) are uc-cones, then the definition of an
open (continuous) linear operator T can be written as in the following simple case: an operator T :
\scrP \rightarrow \scrQ is open (continuous) if there is \beta > 0 such that \beta W \subseteq (T \times T )(U) (resp., (T \times T )(U) \subseteq
\subseteq \beta W ).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
SOME APPLICATIONS OF THE OPEN MAPPING THEOREM IN LOCALLY CONVEX CONES 429
Theorem 2.3. Let (\scrP ,U) and (\scrQ ,\scrW ) be uc-cones. Suppose that T is a linear operator of \scrP
onto a subcone \scrQ 1 of \scrQ such that
T (a) = T (b) implies a+N = b+N (2.2)
for all a, b \in \scrP , where N = \mathrm{k}\mathrm{e}\mathrm{r}T. If T is open and continuous, then T \ast is an open continuous
mapping of (\scrQ \ast ,U\beta (\scrQ \ast ,\scrQ )) onto ((N \times N)\circ ,U\beta (\scrP \ast ,\scrP )). Hence:
(1) if , moreover, T is one-to-one, then T \ast maps onto \scrP \ast ;
(2) if T maps onto \scrQ , then T \ast is an isomorphism of \scrQ \ast onto (N \times N)\circ ;
(3) if T is an isomorphism of \scrP onto \scrQ , then T \ast is an isomorphism of \scrQ \ast onto \scrP \ast .
Proof. Let \mu \in \scrQ \ast be arbitrary. If (x, y) \in N \times N, then \mu (T (x)) = \mu (T (y)) = 0. Thus
T \ast (\mu )(x) \leq T \ast (\mu )(y) + 1.
Hence T \ast (\mu ) is in (N\times N)\circ . Conversely, choose an element f of (N\times N)\circ . We must find an element
\mu of \scrQ \ast such that \mu \circ T = f, that is, the values of \mu on \scrQ 1 must be given by \mu (T (x)) = f(x).
It is, in fact, possible to use this formula to define \mu on \scrQ 1. Observe that if T (x) = T (y), then
by the hypothesis we have x + N = y + N. So x + n = y + n\prime for some n, n\prime \in N. Since f
vanishes on N, f(x) = f(y). (We note that N is a subcone of \scrP and (N \times N)\circ = \{ \mu \in \scrP \ast :
\mu (n) = 0 for all n \in N\} .) Hence \mu is well defined. We show next that the functional \mu is continuous
on \scrQ 1. Since f \in (N \times N)\circ \subseteq \scrP \ast , there exists \alpha > 0 such that f \in (\alpha U)\circ , that is, (x, y) \in \alpha U
implies f(x) \leq f(y) + 1. By openness of T, there is \beta > 0 such that \beta W \subseteq (T \times T )(\alpha U). Now if
a, b \in \scrQ 1 and (a, b) \in \beta W, then there is (a\prime , b\prime ) \in \alpha U such that T (a\prime ) = a and T (b\prime ) = b. We have
f(a\prime ) \leq f(b\prime ) + 1, that is,
(\mu \circ T )(a\prime ) \leq (\mu \circ T )(b\prime ) + 1.
Hence \mu (a) \leq \mu (b) + 1. Therefore \mu \in (\beta W \cap (\scrQ 1 \times \scrQ 1))
\circ and so \mu is continuous on \scrQ 1. By
Theorem 2.2, we can extend \mu to a continuous linear functional on \scrQ . This shows that T \ast is onto
(N \times N)\circ . We will now prove that T \ast is an open mapping. We show that there exists \beta > 0 such
that \beta \widetilde U\circ \subseteq (T \ast \times T \ast )(\widetilde W \circ ). Since T is open, there is \beta > 0 such that \beta W \subseteq (T \times T )(U). Thus
\beta ((T\times T )(U))\circ \subseteq W \circ . By Lemma 2.3 (1), T \ast - 1(U\circ ) = ((T\times T )(U))\circ . Hence \beta T \ast - 1(U\circ ) \subseteq W \circ .
Then \beta T \ast (T \ast - 1(U\circ )) \subseteq T \ast (W \circ ). Since T \ast is surjective, \beta U\circ \subseteq T \ast (W \circ ) and so \beta \widetilde U\circ \subseteq \widetilde T \ast (W \circ ).
By Lemma 2.2, we have \beta \widetilde U\circ \subseteq (T \ast \times T \ast )(\widetilde W \circ ). Thus T \ast is open. Now we show that T \ast is
continuous. We prove that there exists \gamma > 0 such that (T \ast \times T \ast )(\widetilde W \circ ) \subseteq \gamma \widetilde U\circ . Since T is
continuous, there exists \gamma > 0 such that (T \times T )(U) \subseteq \gamma W. Hence W \circ \subseteq \gamma ((T \times T )(U))\circ . By
Lemma 2.3 (1), W \circ \subseteq \gamma T \ast - 1(U\circ ). Thus T \ast (W \circ ) \subseteq \gamma U\circ . Therefore, by Lemma 2.2,
(T \ast \times T \ast )(\widetilde W \circ ) \subseteq \widetilde T \ast (W \circ ) \subseteq \gamma \widetilde U\circ .
Hence T \ast is continuous.
(1) If T is one-to-one, then N = \{ 0\} , so (N \times N)\circ is the whole of \scrP \ast .
(2) If T maps onto \scrQ , then by Lemma 2.1, T \ast is one-to-one. Since it is also open and continu-
ous, it is an isomorphism of \scrQ \ast onto (N \times N)\circ .
(3) The result follows from (1) and (2).
Theorem 2.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
430 S. JAFARIZAD, A. RANJBARI
The above theorem holds for a normed space without condition (2.2) (see, for example, [3],
II.24.4). In the following example (similar to Example 2.2 of [7]) we show that if condition (2.2)
does not hold, then the functional \mu is not necessarily well defined, i.e., the mapping T \ast is not
necessarily onto (N \times N)\circ .
Example 2.1. Let \scrP =
\bigl\{
(x, y) \in \BbbR 2| x, y \geq 0
\bigr\}
, endowed with the convex quasiuniform structure
U = \{ \alpha \widetilde (1, 1) : \alpha > 0\} , where
\widetilde (1, 1) =
\Bigl\{ \bigl(
(a, b), (c, d)
\bigr)
\in \BbbR 2
+ \times \BbbR 2
+ : (a, b) \leq (c, d) + (1, 1)
\Bigr\}
(the order in \BbbR 2
is coordinatewise) and let the subcone \scrQ 1 = \BbbR + = [0,+\infty ] of \scrQ = \BbbR , endowed
with the convex quasiuniform structure \widetilde \scrV = \{ \~\varepsilon : \varepsilon > 0\} , where
\~\varepsilon =
\Bigl\{
(a, b) \in \BbbR 2
+ : a \leq b+ \varepsilon
\Bigr\}
.
Let T be the linear mapping from \scrP onto \scrQ 1 defined by T (x, y) = x + y. Then we have N =
= \mathrm{k}\mathrm{e}\mathrm{r}(T ) = \{ (0, 0)\} . The mapping T : (\scrP ,U) \rightarrow (\scrQ 1, \widetilde \scrV ) is open and continuous. Let f be the
linear functional on \scrP defined by f(x, y) = x. This functional is continuous and is an element of
(N \times N)\circ . But there exists no \mu in \scrQ \ast
1 such that \mu \circ T = f on \scrQ 1. Indeed, the dual cone of (\scrQ 1, \widetilde \scrV )
is the positive reals together with 0 which maps all a \in \BbbR + to 0 and +\infty to +\infty and any of this
functional does not satisfy in the relation \mu \circ T = f. We note that condition (2.2) does not hold for
this T, for example, T (0, 1) = T (1, 0), but (0, 1) +N \not = (1, 0) +N.
Remark 2.1. For topological vector spaces, an operator T is one-to-one if and only if \mathrm{k}\mathrm{e}\mathrm{r}(T ) =
= \{ 0\} , but in locally convex cones, this is not true. In locally convex cones if T is one-to-one, then
\mathrm{k}\mathrm{e}\mathrm{r}(T ) = \{ 0\} , but the converse is not true. For instance, in the Example 2.1 we have \mathrm{k}\mathrm{e}\mathrm{r}(T ) =
= \{ (0, 0)\} , however, T is not one-to-one.
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Received 16.07.18,
after revision — 03.07.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 3
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| id | umjimathkievua-article-222 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:19Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-2222025-03-31T08:48:21Z Some applications of the open mapping theorem in locally convex cones Some applications of the open mapping theorem in locally convex cones Jafarizad, S. Ranjbari, A. Jafarizad, S. Ranjbari, A. Locally convex cones adjoint open mapping uc-cone LATEXstyle Locally convex cones adjoint open mapping uc-cone LATEXstyle UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones.  Specially, we prove some relations between an open linear operator and its adjoint in  $uc$-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).   УДК 515.12Деякi застосування теореми про вiдкрите вiдображення у локально-опуклих конусах Показано, що неперервний відкритий лінійний оператор зберігає повноту та бочкуватість у локально-опуклих конусах.  Зокрема, доведено деякі співвідношення між відкритим лінійним оператором та його суміжним у $uc$-конусах (локально-опуклих конусах, у яких опуклі квазірівномірні структури генеруються одним елементом). Institute of Mathematics, NAS of Ukraine 2021-03-19 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/222 10.37863/umzh.v73i3.222 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 3 (2021); 425 - 430 Український математичний журнал; Том 73 № 3 (2021); 425 - 430 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/222/8985 Copyright (c) 2021 Somayyeh Jafarizad, Asghar Ranjbari |
| spellingShingle | Jafarizad, S. Ranjbari, A. Jafarizad, S. Ranjbari, A. Some applications of the open mapping theorem in locally convex cones |
| title | Some applications of the open mapping theorem in locally convex cones |
| title_alt | Some applications of the open mapping theorem in locally convex cones |
| title_full | Some applications of the open mapping theorem in locally convex cones |
| title_fullStr | Some applications of the open mapping theorem in locally convex cones |
| title_full_unstemmed | Some applications of the open mapping theorem in locally convex cones |
| title_short | Some applications of the open mapping theorem in locally convex cones |
| title_sort | some applications of the open mapping theorem in locally convex cones |
| topic_facet | Locally convex cones adjoint open mapping uc-cone LATEXstyle Locally convex cones adjoint open mapping uc-cone LATEXstyle |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/222 |
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