F1000ResearchF1000Research2046-1402F1000 Research LimitedLondon, UK10.12688/f1000research.174190.2Research ArticleArticlesNew Kind of orthogonality in Normed spaces by using ϼ
* -orthogonality
[version 2; peer review: 3 approved]
Ahmed Jasim - ليث احمد جاسم محمدLaithProject AdministrationWriting – Original Draft PreparationWriting – Review & Editinghttps://orcid.org/0009-0000-6299-7094a1Yahya AbedMohammedConceptualizationMethodologyProject AdministrationSupervisionValidationWriting – Review & Editing2
University of Kerbala, Karbala, Karbala Governorate, Iraq
University of Kerbala, Karbala, Karbala Governorate, Iraqlaith.ahmed@s.uokerbala.edu.iq
Orthogonality is an important concept in normed spaces, and many generalizations of this concept have been proposed. In this paper, we introduce a new concept of orthogonality in normed spaces, denoted by
ϼLM-orthogonality, which is related to
ϼ∗-orthogonality. The fundamental properties of this new concept are systematically analyzed, including symmetry, non-negativity, scalar homogeneity, linear independence of orthogonal vectors, the Cauchy–Schwarz type inequality, the zero property, and other characteristics. We also explain the relationship between this new definition and some other well-known types of orthogonality.
orthogonalitynorm derivativesϼ_*-orthogonality.The author(s) declared that no grants were involved in supporting this work.Amendments from Version 1
Dear Reviewer, Thank you for the reviewers’ valuable comments and suggestions. I would like to inform you that all requested revisions have been carefully addressed in the revised version of the manuscript. In particular: 1. The typographical and formatting errors have been corrected. 2. The motivation behind the proposed orthogonality has been clarified in the Introduction. 3. The proof of Theorem 2.2 has been explained more clearly. 4. Example 3.4 has been revised and clarified. 5. References have been carefully checked and corrected. 6. The manuscript does not contain any tables. 7. A conclusion section has been added to summarize the main contributions of the paper. Thank you again for your time and consideration. I sincerely appreciate the reviewers’ efforts and hope that the revised manuscript will now be suitable for approval. Best regards,
1. Introduction
In an inner product space
(Ẋ,⟨⋅,⋅⟩), two vectors
ṡ,ṙinẊ are said to be orthogonal, denoted by
ṡ⊥ṙif and only if
⟨ṡ,ṙ⟩=0. In spaces whose norm is induced by an inner product, the notion of orthogonality admits a natural and uniquely determined form. Extending this concept to more general normed spaces enable a deeper investigation of their geometric and analytical interrelations, thus offering a broader framework to generalize classical Euclidean ideas to abstract mathematical settings.
Consider
(Ẋ,‖.‖) as a real normed vector space whose dimension is not less than two, there exist several notions of orthogonality, among which the definition introduced by (Birkhoff–James) is one of the most prominent. If
ṡ,ṙ∈Ẋ, define
ṡ⊥Bṙ ⇔
‖ṡ+tṙ‖≥‖ṡ‖, (t
∈R) see.
1
In 1986, Amir in
2 defined the norm derivatives using two functionals,
ϼ−,ϼ+:Ẋ×Ẋ→ℝ be functionals, as follows:
ϼ−(ṡ,ṙ)=‖ṡ‖limt→0−‖ṡ+tṙ‖−‖ṡ‖tandϼ+(ṡ,ṙ)=‖ṡ‖limt→0+‖ṡ+tṙ‖−‖ṡ‖tthen ṡ is
ϼ−- orthogonality to
ṙ denoted by
ṡ⊥ϼ−ṙ⟺ϼ−(ṡ,ṙ)=0 and ṡ is
ϼ+- orthogonality to
ṙ denoted by
ṡ⊥ϼ+ṙ⟺ϼ+(ṡ,ṙ)=0.
A mapping
⟨⋅,⋅⟩g:Ẋ×Ẋ→ℝ was introduced by MiliĆiĆ in
3 as follows:
ϼ(ṡ,ṙ)=⟨ṡ,ṙ⟩g=ϼ−(ṡ,ṙ)+ϼ+(ṡ,ṙ)2. But M. Nur and H. Gunawan in
4 defined an orthogonality relation based on norm derivatives by
ϼgg=|ϼ(ṡ,ṙ)||ϼ(ṙ,ṡ)|, Also Zamani and Moslehian in
5 defined an orthogonality functional
ϼλ::Ẋ×Ẋ→ℝ by
ϼλ(ṡ,
ṙ) = λ
ϼ−(ṡ,ṙ) + (1- λ)
ϼ+(ṡ,ṙ) ∀ λ
∈R, and ṡ is
ϼλ- orthogonality to
ṙ denoted by
ṡ⊥ϼλṙ⟺ϼλ(ṡ,ṙ)=0 the notions of
ϼ∗-orthogonality is introduced in
6 as:
ṡ⊥ϼ∗ṙ⟺ϼ∗(ṡ,ṙ)=ϼ+(ṡ,ṙ)ϼ_(ṡ,ṙ)= 0. Recall that
Ẋ is smooth at
ṡif and only if
ϼ_(ṡ,ṙ)=ϼ+(ṡ,ṙ)for all
ṙ∈Ẋ.
These functionals extend the concept of the inner product to a normed space, and many geometric properties will be reformulated in the normed space using the concept of the norm derivative. Now we shall address some important theorem that we will need in this research.
6Suppose (
Ẋ, ‖·‖) represents a real normed vector space, then
ϼ∗(t ṡ,
ṙ) =
ϼ∗(ṡ, t
ṙ)=
t2ϼ∗(ṡ,
ṙ) for all ṡ,
ṙ ∈
Ẋ and all t ∈ R.
|
ϼ∗(ṡ,ṙ)|
≤‖ṡ‖2‖ṙ‖2 for all ṡ,
ṙ∈Ẋ.
For all vectors except the zero vector, ṡ,
ṙ ∈
Ẋ, if
ṡ⊥ϼ∗ṙ, then ṡ and
ṙ are independent in the linear sense.
ϼ∗(ṡ, t ṡ +
ṙ) =
t2‖ṡ‖4+2t
‖ṡ‖2ϼ(ṡ,
ṙ) +
ϼ∗(ṡ,
ṙ) for all ṡ,
ṙ∈Ẋ and all t ∈ R.
For more information about other orthogonalities you can see.
7–
10
2. Fundamental results
This section is devoted to introducing a new form of orthogonality by using
ϼ∗-orthogonality as follows:
Suppose (
Ẋ, ‖·‖) represents a real normed vector space then
∀ṡ,ṙ∈Ẋ, the functional
ϼLM:Ẋ×Ẋ→Rwhich defines as
ϼLM(ṡ,ṙ)=|ϼ∗(ṡ,ṙ)ϼ∗(ṙ,ṡ)|, We say that ṡ is
ϼLM- orthogonality to
ṙ, we denoted by ṡ
⊥ϼLMṙ if
ϼLM(ṡ,ṙ)=0.
The notion of
ϼLM-orthogonality is motivated by the need for a symmetric orthogonality relation built upon
ϼ*-orthogonality. Since
ϼ*-orthogonality is not symmetric in general, we introduce a symmetric and non-negative functional that retains the core geometric structure of the original relation, thereby providing a more balanced framework for comparing vectors in normed spaces.
Suppose (
Ẋ, ‖·‖) represents a real normed vector space. Then
ϼLM(ṡ,ṙ)≥0 for all ṡ,
ṙ∈Ẋ. (non-negative)
ϼLM(ṡ,0)=ϼLM(0,ṙ)=0for all ṡ,
ṙ∈Ẋ. (zero property)
ϼLM(ṡ,ṡ)=‖ṡ‖4 for all ṡ
∈Ẋ, also
ϼLM(ṡ,ṡ) =
ϼ∗(ṡ,ṡ)
ϼLM(ṡ,ṙ)=ϼLM(ṙ,ṡ) for all ṡ,
ṙ∈Ẋ. (symmetry)
ϼLM(αṡ,βṙ)=|αβ|2ϼLM(ṡ,ṙ) for all
ṡ,ṙ∈Ẋand all
α,β∈R. (scalar homogeneity)
ϼLM(ṡ,ṙ)≤‖ṡ‖2‖ṙ‖2 for all ṡ,
ṙ∈Ẋ. (Cauchy schwarz inequality)
By theorem 1.1 (ii) we have |
ϼ∗(ṡ,ṙ))|
≤‖ṡ‖2‖ṙ‖2 for all
ṡ,ṙ∈Ẋ
Then
ϼLM(ṡ,ṙ)=|ϼ∗(ṡ,ṙ)ϼ∗(ṙ,ṡ)|≤‖ṡ‖2‖ṙ‖2‖ṙ‖2‖ṡ‖2=
‖ṡ‖2‖ṙ‖2 for all
ṡ,ṙ∈Ẋ
Hence
ϼLM(ṡ,ṙ)≤‖ṡ‖2‖ṙ‖2 for all ṡ,
ṙ∈Ẋ
By theorem 1.1 (iv) we have
ϼ∗(ṡ,tṡ+ṙ) =
t2‖ṡ‖4+2t
‖ṡ‖2ϼ(ṡ,ṙ) +
ϼ∗(ṡ,ṙ)ϼLM(ṡ,tṡ+ṙ)=|ϼ∗(ṡ,tṡ+ṙ)ϼ∗(tṡ+ṙ,ṡ)|=|(t2‖ṡ‖4+2t‖ṡ‖2ϼ(ṡ,ṙ)+ϼ∗(ṡ,ṙ))(ϼ∗(tṡ+ṙ,ṡ))|
we have ṡ
⊥ϼLMṙ therefore;
ϼLM(ṡ,ṙ)=|ϼ∗(ṡ,ṙ)ϼ∗(ṙ,ṡ)| = 0
Thus
ϼ∗(ṡ,ṙ) = 0 or
ϼ∗(ṙ,ṡ) = 0
By theorem 1.1 (iii) for all nonzero vectors ṡ,
ṙ∈Ẋ, if
ṡ⊥ϼ∗ṙ, then ṡ and
ṙ are linearly independent.
Hence, whether
ϼ∗(ṡ,ṙ) = 0 or
ϼ∗(ṙ,ṡ) = 0 it follows that ṡ and
ṙ are linearly independent.
If
Ẋ come from inner product space then
ϼLM(ṡ,ṙ) =
ϼ∗(ṡ,ṙ).
3. The connection between
<inline-formula>
<mml:math display="inline">
<mml:msub>
<mml:mi mathvariant="normal">ϼ</mml:mi>
<mml:mi mathvariant="bold-italic">LM</mml:mi>
</mml:msub>
</mml:math>
</inline-formula> – orthogonality and some others orthogonalities
Suppose
(Ẋ,‖·‖) represents a real normed vector space and let
ṡ,
ṙ ∈
Ẋ.
If
ṡ⊥ϼ∗ṙ, then
ṡ⊥ϼLMṙ
If
ṡ⊥ϼ−ṙ, then
ṡ⊥ϼLMṙ
If
ṡ⊥ϼ+ṙ, then
ṡ⊥ϼLMṙ
If
ṙ⊥ϼ∗ṡ, then
ṡ⊥ϼLMṙ
If
ṙ⊥ϼ−ṡ, then
ṡ⊥ϼLMṙ
If
ṙ⊥ϼ+ṡ, then
ṡ⊥ϼLMṙ
Obvious
The converse of (a), (b) and (c) of theorem 3.1 are not true. For example, take
ṡ=(0,2) and
ṙ=(1,1) if
‖(ṡ1,ṡ2)‖=max{|ṡ1|,|ṡ2|}in the space
Ẋ =
R2ϼ+(ṡ,ṙ)=‖ṡ‖limt→0+‖ṡ+tṙ‖−‖ṡ‖t=(2)limt→0+|2+t|−2t=2(1)=2≠0ϼ−(ṡ,ṙ)=‖ṡ‖limt→0−‖ṡ+tṙ‖−‖ṡ‖t=(2)limt→0−|2+t|−2t=2(1)=2≠0ϼ+(ṙ,ṡ)=‖ṙ‖limt→0+‖ṙ+tṡ‖−‖ṙ‖t=(1)limt→0+|1+2t|−1t=1(2)=2ϼ−(ṙ,ṡ)=‖ṙ‖limt→0−‖ṙ+tṡ‖−‖ṙ‖t=(1)limt→0−|1|−1t=
Since
ϼ∗(ṡ,ṙ)=ϼ+(ṡ,ṙ)ϼ_(ṡ,ṙ)ϼ∗(ṡ,ṙ)=(2)(2)=4≠0
But
ϼLM(ṡ,ṙ)=|ϼ∗(ṡ,ṙ)ϼ∗(ṙ,ṡ)| = 0
Hence ṡ
⊥ϼLMṙbut the others are not.
The converse of (d), (e) and (f
) of theorem 3.1 are not true. For example, take ṡ = (2,0,0) and
ṙ = (1,1,0) if
‖(ṡ1,ṡ2,ṡ3)‖ =
|ṡ1|+|ṡ2| +
|ṡ3| in the space X =
R3, then
ϼ+(ṡ,ṙ)=‖ṡ‖limt→0+‖ṡ+tṙ‖−‖ṡ‖t=(2)limt→0+|2+t|+|t|−2t=2(2)=4ϼ−(ṡ,ṙ)=‖ṡ‖limt→0−‖ṡ+tṙ‖−‖ṡ‖t=(2)limt→0−|2+t|+|t|−2t=2(0)=0ϼ+(ṙ,ṡ)=‖ṙ‖limt→0+‖ṙ+tṡ‖−‖ṙ‖t=(2)limt→0+|1+2t|+|1|−2t=2(2)=4≠0ϼ−(ṙ,ṡ)=‖ṙ‖limt→0−‖ṙ+tṡ‖−‖ṙ‖t=(2)limt→0−|1+2t|+|1|−2t=2(2)=4≠0
Since
ϼ∗(ṙ,ṡ)=ϼ+(ṙ,ṡ)ϼ−(ṙ,ṡ)ϼ∗(ṙ,ṡ)=(4)(4)=16≠0
But
ϼLM(ṡ,ṙ)=|ϼ∗(ṡ,ṙ)ϼ∗(ṙ,ṡ)| = 0
Hence ṡ
⊥ϼLMṙbut the others are not.
Consider the real normed space
Ẋ =
R2 with the norm
‖ṡ‖ =
|ṡ1|+|ṡ2| where
ṡ=(1,0),
ṙ=(0,1) we have
ϼ+(ṡ,ṙ) =
‖ṡ‖limt→0+‖ṡ+tṙ‖−‖ṡ‖t and
ϼ−(ṡ,ṙ) =
‖ṡ‖limt→0−‖ṡ+tṙ‖−‖ṡ‖t
‖ṡ‖=1,‖ṙ‖=1,
‖ṡ+tṙ‖=‖(1,t)‖=|1|+|t|
and
‖ṙ+tṡ‖=‖(t,1)‖=|t|+|1|
Thus
ϼ+(ṡ,ṙ) = (1)
limt→0+1+t−1t = 1
ϼ−(ṡ,ṙ)=
(1)limt→0−1−t−1t = -1
ϼ+(ṙ,ṡ) = (1)
limt→0+t+1−1t = 1
ϼ−(ṙ,ṡ)=
(1)limt→0−−t+1−1t= -1
We have
ϼ(ṡ,ṙ) =
ϼ−(ṡ,ṙ)+ϼ+(ṡ,ṙ)2 therefore
ϼ(ṡ,ṙ) =
−1+12 = 0,
Since
ϼgg=|ϼ(ṡ,ṙ)||ϼ(ṙ,ṡ)|implies
ϼgg(ṡ,ṙ) =
0 = 0,
Consider the real normed space
Ẋ =
R2 with the norm
‖(ṡ,ṙ)‖ = max {
|ṡ|,|ṙ|} where
ṡ=(1,1),
ṙ=(0,−1). Then for every
ṡ,ṙ∈Ẋ we have
ϼ+(ṡ,ṙ)=(1)limt→0+|1|−1t=1(0)=0ϼ−(ṡ,ṙ)=(1)limt→0−|1−t|−1t=1(−1)=−1ϼ+(ṙ,ṡ)=(1)limt→0+|t|−1t=1(0)=0ϼ−(ṙ,ṡ)=(1)limt→0−|−1+t|−1t=limt→0−|t−1|−1t=−1
We have
ϼ(ṡ,ṙ) =
ϼ−(ṡ,ṙ)+ϼ+(ṡ,ṙ)2 therefore
ϼ(ṡ,ṙ) =
−1+02 =
−12≠0 and
ϼ(ṙ,ṡ) =
ϼ−(ṙ,ṡ)+ϼ+(ṙ,ṡ)2 =
−1+02 =
−12ϼgg(ṡ,ṙ)=(−12)(−12)=(14)=12≠0ϼλ(ṡ,ṙ)=λϼ−(ṡ,ṙ)+(1−λ)ϼ+(ṡ,ṙ))=λ(−1)+(1−λ)(0)=−λ≠0but
ϼLM(ṡ,
ṙ) =
|ϼ∗(ṡ,ṙ)ϼ∗(ṙ,ṡ)|= 0.
Hence
⊥ϼLM⊈⊥ϼ,
⊥ϼLM⊈⊥ϼggand
⊥ϼLM⊈⊥ϼλ
From example 3.4 and example 3.5 it follows that the orthogonalities,
⊥ϼ,
⊥ϼgg,
⊥ϼλand
⊥ϼLM are generally incomparable.
If
X is smooth then
⊥ϼ−=⊥ϼ+=⊥ϼ=⊥ϼ∗=⊥ϼλ but not equal to
⊥ϼLM.
4. Conclusion
This paper presented a new orthogonality relation in normed spaces, termed
ϼLM-orthogonality, introduced via
ϼ*-orthogonality. Several fundamental properties were established, among them symmetry, non-negativity, homogeneity, and a Cauchy–Schwarz type inequality.
The connections between
ϼLM-orthogonality and other classical orthogonality concepts were examined, revealing that these notions are generally non-equivalent. Collectively, the results confirm that
ϼLM-orthogonality constitutes a meaningful and structurally rich extension of existing frameworks.
Possible directions for future work include characterizing linear mappings that preserve
ϼLM-orthogonality and investigating its role in the geometry of normed spaces.
Ethical considerations
This study is entirely theoretical and does not involve any human participants or animals, nor does it rely on data requiring ethical approval. Therefore, no ethical concerns arise in conducting this research.
Data availability
This study is entirely theoretical and does not rely on any experimental or external datasets. Therefore, no data are available in connection with this research.
Reporting guidelines
This article is a purely theoretical study in functional analysis and does not involve clinical trials, animal studies, observational studies, or qualitative research. Therefore, standard reporting guidelines such as CONSORT, ARRIVE, STROBE, or COREQ/SRQR do not directly apply. Standard practices for presenting theoretical mathematical results have been followed, including clear definitions, theorems, proofs, and logical consistency.
ReferencesBirkhoffG:
Orthogonality in linear metric spaces.Duke Mathematical Journal.1935;1(2):169–172.
10.1215/S0012-7094-35-00115-6AmirD:
Characterizations of Inner Products Spaces.Basel:
Birkauser Verlag;1986.MiličićPM:
Sur la G-orthogonalité dans les espaces normés.Mat. Vesn.1987;39:325–334.NurMGunawanH:
A new orthogonality and angle in a normed space.Aequationes Mathematicae.2019;93(3):547–555.ZamaniAMoslehianMS:
An extension of orthogonality relations based on norm derivatives.The Quarterly Journal of Mathematics.2019;70(2):379–393.MoslehianMSZamaniADehghaniM:
Characterizations of smooth spaces by
ϼ∗-orthogonality.Houston Journal of Mathematics.2017;43(4):1061–1074.ChmielińskiJWójcikP:
On a
ϼ-orthogonality.Aequationes Math.2010;80(1):45–55.
10.1007/s00010-010-0042-1AbedMY:
New form of orthogonality in the real normed spaces.Journal of Interdisciplinary Mathematics.2022;25(8):2579–2584.
10.1080/09720502.2022.2057047AbedMY:
On semi and quasi
γϼϼ-orthogonality in real normed space.Journal of Interdisciplinary Mathematics.2022;25(8):2585–2590.
10.1080/09720502.2022.2057048AlonsoJBenítezC:
Orthogonality in normed linear spaces: A survey. Part II: Relations between main orthogonalities.Extracta Mathematicae.1989;4(3):121–131.10.5256/f1000research.201643.r488721Reviewer response for version 2IshtiaqUmar1Refereehttps://orcid.org/0000-0002-5228-1073
University of Management and Technology, Lahore, Pakistan
Competing interests: No competing interests were disclosed.
The authors revised the manuscript according to my suggestions.
Is the work clearly and accurately presented and does it cite the current literature?
No
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
No
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Functional Analysis
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
10.5256/f1000research.192072.r477500Reviewer response for version 1GnanaprakasamArul Joseph1Referee
SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu, India
Competing interests: No competing interests were disclosed.
The authors introduced a new notion of orthogonality based on norm derivatives and established several interesting properties related to this concept. The proposed orthogonality is discussed in comparison with existing notions in the literature, and the paper provides useful remarks and illustrative examples to clarify the results.
The manuscript is mathematically sound and well organized. The topic is relevant to current research in normed spaces and orthogonality theory, and the obtained results may motivate further studies in this direction. Overall, the paper makes a valuable contribution to the subject and is suitable for indexingin its present form.
Is the work clearly and accurately presented and does it cite the current literature?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Partly
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Partly
Reviewer Expertise:
Functional Analysis
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
10.5256/f1000research.192072.r468033Reviewer response for version 1GhoshSouvik1Referee
Jadavpur University, Kolkata, West Bengal, India
Competing interests: No competing interests were disclosed.
The authors tried to provide a new orthogonality based on norm derivatives. The orthogonality here mentioned is coarser than other orthogonalities in the literature. Although they showed that it is incomparable with some other orthogonalities followed by some remarks. The authors may look for the following problems for further exploration. For instance by investigating the quantitative invariant other gap between two different (incomparable) orthogonalities. Also, there is a scope to study the preservation of such orthogonality.
Is the work clearly and accurately presented and does it cite the current literature?
No
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Partly
Are the conclusions drawn adequately supported by the results?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
No
Reviewer Expertise:
Geometry of Banach spaces (Functional Analysis)
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
10.5256/f1000research.192072.r458032Reviewer response for version 1IshtiaqUmar1Refereehttps://orcid.org/0000-0002-5228-1073
University of Management and Technology, Lahore, Pakistan
Competing interests: No competing interests were disclosed.
The manuscript contains several typographical errors and formatting inconsistencies, such as repeated sentences in the introduction and missing mathematical symbols in Theorem 1.1.
The definition of $\rho_{LM}$-orthogonality in Definition 2.1 is clear, but the motivation for introducing this specific form of orthogonality could be better justified.
The proof of Proposition 2.2 contains multiple formatting issues and missing parentheses, making some steps difficult to follow.
In Example 3.4, the calculation of $\rho_{LM}(\dot{s},\dot{t})$ appears to contain an error, as the result should be verified more carefully.
The references are inconsistently formatted, with some entries missing page numbers or having incomplete information (e.g., references 4, 5, and 6).
The paper references figures and tables that are not included in the provided PDF, making it difficult to verify the claims made in examples.
The paper lacks a formal conclusion section to summarize the main findings and their significance in the context of existing orthogonality concepts.
Is the work clearly and accurately presented and does it cite the current literature?
No
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
No
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Functional Analysis
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.