On Orthogonality according to an Index in Semi–normed Spaces
ARTUR STRINGA, ARBËR QOSHJA
Department of Mathematics, Department of Applied Mathematics,
University of Tirana,
Sheshi “Nënë Tereza”, Tirana,
ALBANIA
Abstract: - In a recent study, we introduce the concept of orthogonality and transversality according to an
index, obtaining some results on linear dependence and independence in semi-normed spaces. In this paper, we
discuss the concept of orthogonality in semi-normed spaces.
Key-Words: semi-norm, semi-pre-inner product, orthogonality according to an index, orthogonality
Received: December 15, 2022. Revised: June 2, 2023. Accepted: June 23, 2023. Published: July 24, 2023.
1 Introduction
This paper derives from our previous work, [1].
Specifically, to carry over Hilbert space type
arguments to the theory of Banach spaces, [3],
constructed on a vector space a type of inner
product, named semi–inner product (s.i.p), [9], with
a more general axiom system that of Hilbert space,
[4].
Definition 1.1
Let X be a real vector space. We say that a real
semi-inner product (in short s.i.p.) is defined on
X if for every there corresponds a real number
and the following properties hold, [3]:
(1)(i)
(ii) for and
(2) for
(3)
The pair is a semi-inner product space (in
short, s.i.p.s.).
A s.i.p.s. is a normed vector space with
, [3].
For Lumer the importance of an s.i.p. space is
that every normed vector space can be represented
as an s.i.p. space so that the theory of operators on
Banach space can be penetrated by Hilbert space
arguments, [4].
Aiming to generalize condition (2) in the
definition of s.i.p., we have introduced in [1], the
semi-pre-inner product function, which is a
generalization of the s.i.p. function’s concept.
Definition 1.2
Let X be a real vector space. Consider a function
defined on as follows, [2]:
.
If satisfies the conditions:
(1)
(2) and
(3)
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(4)
then, we say that is a semi-pre-inner product on X
(in short s.p.i.p.).
The pair is called a semi-pre-inner product
space (in short s.p.i.p.s.).
In [2], it proved that for every semi-norm function p
in the vector space X, there is a s.p.i.p. , such
that
Let X be a real vector space and be a
semi-normed space, where is a family of
semi-norms on X and is an index set. For every
let us denote by the s.p.i.p.,
corresponding to the semi-norm .
In [1], we defined an orthogonality relation in
s.p.i.p. spaces as follows:
Definition 1.3
Let be and . The vector x is called
orthogonal according to the index over the vector
y, if . In this case, the vector y is called
transversal according to the index over the vector
x, [1].
Definition 1.4
Let be . The vector x is called orthogonal
over the vector y, if the vector x is orthogonal
according to every index over the vector y.
In this case, the vector y is called transversal over
the vector x, [1].
Definition 1.5
Let
,V
be a normed linear space. Denote by S
the unit sphere in V. The normed space
,V
is
called Gâteaux differentiable [4], if for all
,x y S
and real
:
0
lim
x y x
exists.
Definition 1.6
Let
X
be a s.i.p. space. Denote by S the unit
sphere in X. The s.i.p. space
X
is called a
continuous s.i.p. space if for all
‚x y S
and real
, [4]:
0‚‚
lim y x y y x .
Theorem 1.1
An s.i.p. space is a continuous s.i.p. space if and
only if the norm is Gâteaux differentiable, [4].
Theorem 1.2
In a continuous s.i.p. space
X
x is normal to
, which is equivalent to
0 xy
, if and only if
x y x
for all scalar
, [4].
2 Main Results
Our first main concern is to define the class of
continuous s.p.i.p. spaces. We will show that in such
spaces the orthogonality [5] according to an index is
a generalization to the orthogonality relation as
studied by J. Gilles in Theorem 1.2. Also, we show
that in continuous s.p.i.p. spaces, it holds similar
results compared with the results of Theorem 1.1.
In the end, we will get some good results for
orthogonality according to an index on separable
semi-normed spaces, [6].
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Definition 2.1
Let be an s.p.i.p. space and p the semi-
norm, corresponding to the s.p.i.p. .
The s.p.i.p. space is called a continuous
s.p.i.p. space if the s.p.i.p. has the property:
For every such that
1,p x p y
Let X be a real vector space and be a
semi-normed space, where is a family of
semi-norms on X and is an index set. For every
let us denote by the s.p.i.p.,
corresponding to the semi-norm .
Theorem 2.1
Let be , such that the s.p.i.p. space
is a continuous s.p.i.p. space.
Let be such that
The vector x is orthogonal according to the index
over the vector y, if and only if for every
.
Proof: Let be , such that the s.p.i.p. space
is a continuous s.p.i.p. space.
Let be such that
Assume that for every
,
R
.
For all, it’s true that
On the other hand, using (3) and (4) properties of
semi–pre–inner product, for every we have:
From here it follows that:
if then and if then
It’s true that:
From the above equations, the following inequalities
hold:
and
So, As a result, the vector is
orthogonal according to the index over the vector
.
Assume that the vector is orthogonal according to
the index over the vector . For every, we
have that:
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from which, forever , it’s held the inequality
.
Corollary 2.1
Let's assume that for every , the s.p.i.p. space
is a continuous s.p.i.p. space.
Let be such that for every index
,
The vector x is orthogonal over the vector y, if and
only if for every index and for every
.
Theorem 2.2
Let be . The s.p.i.p. space is a
continuous s.p.i.p. space if and only if for every
such that
1p x p y
:
0
lim
p x y p x
exists.
Proof: Let be .
Assume that the s.p.i.p. space is a
continuous s.p.i.p. space.
Since is a semi–norm on X, we have that
is a closed subspace
of the vector space X. We note that the relation:
is an equivalent relation in X. Let denote by the
quotient set with respect to this
equivalence relation and by an equivalence class
with a representative x. The function:
such that for every
is a norm in , [2]. Then, by [3], there exists a
s.i.p. on :
such that for every
Let us consider the function:
such that
for every
The above function is a s.p.i.p. function, [2].
Let be such that so
Since:
by [4], we have that the norm function
is Gâteaux differentiable, i.e.,
exists.
On the other hand, since for every the
following equations are true:
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and
,
it follows that it exists
Let us assume now that exists
for every two points such that
From here, it follows that
exists , for every two points
such that So, the
norm function is Gâteaux
differentiable. By [4], we have that the s.i.p. space
is a continuous s.i.p. space, from
where it follows that
On the other hand, since for all hold
equations:
and
we get that for every two points such that
Thus, the s.p.i.p. space is a continuous
s.p.i.p. space.
As the semi–normed space can be
considered filtered, [2], we have that
where , if for all
Definition 2.2
Let be , and . The vector x is
called orthogonal according to the index over the
set , if the vector is orthogonal according to the
index overall vectors .
In this case, the set is called transversality, [7],
[8], according to the index over the vector .
Definition 2.3
Let be and . The vector is called
orthogonal over the set if the vector is
orthogonal according to every index over
the set . In this case, the set is called
transversality over the vector .
Definition 2.4
Let be and . The set is called
orthogonal according to the index over the set ,
if all vectors are orthogonal according to the
index over the set In this case, the set is
called transversality according to the index over
the set .
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Definition 2.5
Let be . The set is called orthogonal over
the set , if the set is orthogonal according to
every index over the set . In this case, the
set is called transversality over the set .
Let be and Denote by the set of
all vectors , which are orthogonal according to
the index over the set and denote by the set
of all vectors , which are orthogonal over the
set . It is clear that .
Theorem 2.3
Let's assume that semi–normed space
is separable, i.e., we assume that this set satisfies the
condition:
Let be and then there is an index
, such that
Let be and The following inclusions
are true:
(i)
(ii) ;
Proof: (i) Let be Then,
From here it follows that thus
(ii) Let be Then, for all index ,
From (i) it follows that for every index
Since the semi–normed
space is separable, we get So,
On the other hand, it is clear
We note that if B is a linear subset of X, then
Theorem 2.4
Let be a separable semi–normed
space.
The following statements are equivalent:
(i) If for points there is a , such
that then for every , we have
that .
(ii) If for two points , where there is
only one such that for every , we
have that .
Proof: (i) (ii)
Let be , where Since the semi–
normed space is separable, there is a
, such that Denote
We have that:
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Thus, exists , such that
Since the statement (i) is true, we take that for every
,
Let's suppose that except the exists also
another scalar such that for every ,
In particular:
.
We have:
so, the scalar is unique.
(ii) (i)
Let's assume that for point exists an
, such that
If , then for all ,
Let's suppose that Since the semi–normed
space is separable, then exists a
, such that On the other hand,
since the statement (ii) is true, we get that exists a
unique scalar such that for every
, In particular,
We have:
For all , we have
Theorem 2.5
Let be a separable semi–normed
space.
The following statements are equivalent:
(i) For every two points , there
exists a unique scalar such that for every
, we have that .
(ii) For every two points and for every two
indexes , have:
.
Proof: (i) (ii)
Let be , and , .
If then it is clear
.
Let's suppose Since the semi–normed space
is separable, then exists ,
such that On the other hand, since the
statement (i) is true, we get that exists a unique
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scalar such that for every , we have
For all , we get:
We have:
and
If then from where it
follows:
.
Let suppose that We will prove that for
every , Assume the contrary, so
we assume that exists an , such that
Since the statement (i) is true, based
on Theorem 2.3, we will get that for every ,
In particular, , so
which contradicts the assumption Thus, for
every , From here it follows
that and . Moreover, we
have that:
and .
Dividing side by side the equations (1) and (2) we
will get:
2
2
,
,
xy py
xy py
.
(ii) (i)
Let be , where Assume that exists
, such that .
Since the statement (ii) is true, for all and
, we have:
,
from where it follows for every
and , i.e., for all
.
By Theorem 2.4 we get that exists a unique scalar
such that for all ,
3 Conclusions
This research grew out of our earlier work and our
goal was to see what new advancements may be
achieved for classes of s.p.i.p. spaces formed by
placing additional constraints on the s.p.i.p.
function. We defined the class of continuous s.p.i.p.
spaces. We demonstrated that orthogonality
according to an index in such spaces is a
continuation of the orthogonality relation studied by
J. Gilles. We also demonstrated that the continuity
limitation on the s.p.i.p. function is equal to the
norm's Gâteaux differentiability. In continuous
s.p.i.p. spaces, we obtained similar findings to other
outcomes. Finally, using an index on separable
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semi-normed spaces, we confirmed robust
orthogonality discoveries.
References:
[1] Stringa A., Qoshja A., (2022) On Semi–
Normed Spaces, via Semi–Pre–Inner Product
and Orthogonality According to an Index,
Journal of Multidisciplinary Engineering
Science and Technology (JMEST), Vol. 9,
Issue 12, December 2022, ISSN: 2458-9403.
[2] Stringa A., (2021) On Strictly Convex and
Strictly Convex according to an index Semi-
Normed Vector Spaces, General Mathematics
Notes, Vol. 4, No 2, June 2011, pp.10–22,
ISSN 2219–7184, Copyright ICSRS
Publication, 2011, www.i-crs.org Available
free online at http://www.geman.in.
[3] Lumer G., (1961) Semi-Inner-Product Spaces,
Transactions of the American Mathematical
Society, Vol. 100, No 1, 1961, pp. 29-43.
[4] Giles J.R., (1967) Classes of Semi-Inner-
Product Spaces, Transactions of the American
Mathematical Society, Volume 129, Number 3,
1967, pp. 436-446.
[5] J. A. Wheeler; C. Misner; K.S. Thorne (1973).
Gravitation. W.H. Freeman & Co. p. 58. ISBN
0-7167-0344-0.
[6] Simon, J. (2017). Semi-normed Spaces. In
Banach, Fréchet, Hilbert and Neumann Spaces,
J.Simon (Ed.).
[7] R. Thom, "Un lemma sur les applications
différentiables" Bol. Soc. Mat. Mex. , 1 (1956)
pp. 59–71.
[8] S. Lang, "Introduction to differentiable
manifolds" , Interscience (1967).
[9] J. B. Conway. A Course in Functional
Analysis. 2nd Edition, Springer-Verlag, New
York, 1990, page 1.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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