# Fuzzy Inner Product Spaces And Fuzzy Orthogonality

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### Some aspects of 2-fuzzy inner product space

the 2-fuzzy inner product space is deﬂned. Some important theorems are proved. Moreover crisp inner product and orthogonality are established. Some equivalent conditions are also proved. 2010 AMS Classiﬂcation: 46S40, 03E72 Keywords: 2-Fuzzy inner product space, ﬁ-norm, Fuzzy norm, Crisp inner prod-uct, Orthogonality.

### Walchand College Of Engineering, Sangli.

Vector dot product, Inner product space, Length and orthogonality, Orthogonal sets, Orthonormal sets, Orthogonal projections, Gram-Schmidt Process, Least square problems, A plications and significance of Eigen values and Eigen vectors. Module 3 Fuzzy Sets 7 Hrs. 7 Hrs.

### Walchand College Of Engineering, Sangli.

Vector dot product, Inner product space, Length and orthogonality, Orthogonal sets, Orthonormal sets, orthogonal projections, Gram-Schmidt Process, Least square problems, A lications and Significance ofEi en values and Ei en vectors. Module 3 Fuzzy Sets Introduction, Fuzzy logic, Characteristics functions, Operations on Fuzzy sets, Convex Fuzzy

### RESEARCH Open Access Stability of a generalized quadratic

The pair (X, ⊥)iscalledanorthogonalityspace.By an orthogonality normed space we mean an orthogonality space having a normed structure. Some interesting examples are (i) The trivial orthogonality on a vector spaceX defined by (O 1), and for non-zero (ii) The ordinary orthogonality on an inner product space (X, 〈., 〉)givenbyx ⊥ y if

### Field theories on spaces with linear fuzziness

so called fuzzy sphere , where an irreducible representation of the position operators is used which makes the Casimir of the algebra, (ˆx 1)2 +(ˆx 2)2 +(ˆx 3)2, a multiple of the identity operator (a constant, hence the name sphere). One can consider the square root of this Casimir as the radius of the fuzzy sphere.

### University MSc Tech, Mathematics

5. Inner Product Spaces : Inner Product Spaces, Orthogonality, The Adjoint of Linear Transformation, Unitary operators, Self Adjoints and Normal Operators, Polar and Singular Value Decomposition 6. Bilinear Forms : Definition and Examples, The matrix of a Bilinear Form, Orthogonality, Classification of Bilinear Forms

### Hyers-Ulam-Rassias Stability of Orthogonal Quadratic

ular spaces, that-besides being Banach spaces (or F spaces in more general setting)- are equipped with modular equivalent of norm or metric notions. There are several orthogonality notions on a real normed spaces as Birkho - James, semi-inner product, Carlsson, Singer, Roberts, Pythagorean, isosceles and Diminnie (see, e.g., ).

### COURSE LIST Faculty of Arts & Sciences Department of Mathematics

Inner product and Hilbert spaces Inner product and Hilbert spaces Orthogonality in inner product space, Name of the course in English Fuzzy Numbers and Fuzzy

### Faculty School of Technology Field of study Technomathematics

Content Vector spaces and linear operators. Linear subspaces and projection. Norms, metric and convergence. Function spaces. Banach spaces, Lp-spaces. Inner product and orthogonality. Hilbert spaces. Theory of linear operators, eigenvalues and spectral decomposition. Introduction to wavelet analysis.

### Inner Product Spaces and Orthogonality

this special inner product (dot product) is called the Euclidean n-space, and the dot product is called the standard inner product on Rn. Example 3.2. The vector space C[a;b] of all real-valued continuous functions on a closed interval [a;b] is an inner product space, whose inner product is deﬂned by › f;g ﬁ = Z b a f(t)g(t)dt; f;g 2 C[a

### CURRICULUM VITAE

 E Mostofian, M Azhini, A Bodaghi, Fuzzy inner product spaces and fuzzy orthogonality, Tbilisi Mathematical Journal, 2017, 10 (2), 157-171 (ISI)  Hossein Monfared, Mehdi Asadi, Mahdi Azhini and Donal O Regan, F(ψ,ϕ)-Contractions for α-admissible mappings on M-metric spaces, Fixed Point Theory and Applications, 2018 (1), 22

### Discrete Structures Course Title: Full Marks: 60 + 20 + 20

Unit 8: Orthogonality and Least Squares (5 Hrs.) Inner product, Length, and orthoganility, Orthogonal sets, Orthogonal projections, The Gram-Schmidt process, Least squares problems, Application to linear models, Inner product spaces, Applications of inner product spaces Unit 9: Groups and Subgroups (5 Hrs.)

### INSTRUMENTATION AND CONTROL ENGINEERING

Linear Algebra and Matrices Vector spaces, Subspaces, Basis and Dimension - Systems of Linear Equations. Linear Transformations - Kernel and Image - Geometric Ideas - Inner Product spaces - Orthogonality - Orthonormal basis - Reflections and Orthogonal maps of the plane - Orthogonal complements and Projections.

### Linear algebra 4th edition book - gixefizavemo.weebly.com

Orthogonality and Least Squares Introductory Example: Readjusting the North American Datum 6.1 Inner Product, Length, and Orthogonality 6.2 Orthogonal Sets 6.3 Orthogonal Projections 6.4 The Gram Schmidt Process 6.5 Least-Squares Problems 6.6 Applications to Linear Models 6.7 Inner Product Spaces 6.8 Applications of Inner Product Spaces

### Inner Product Spaces - UC Davis

this section we discuss inner product spaces, which are vector spaces with an inner product deﬁned on them, which allow us to introduce the notion of length (or norm) of vectors and concepts such as orthogonality. 1 Inner product In this section V is a ﬁnite-dimensional, nonzero vector space over F. Deﬁnition 1. An inner product on V is a map

### SOME CHARACTERIZATIONS OF INNER-PRODUCT SPACES

SOME CHARACTERIZATIONS OF INNER-PRODUCT SPACES BY MAHLON M. DAY 1. Introduction. The theorems of this paper give a number of conditions under which the norm in a real-linear or complex-linear normed space can be

### M.TECH. INDUSTRIAL INTELLIGENT SYSTEMS

Vector Spaces: General vector spaces - Sub spaces - Linear independence - Basis Dimension Row space, Column space and Null Space, Eigen values and Eigen vectors. Inner Product Spaces: Inner products - Orthogonality - Orthogonal basis Orthogonal complements - Projection on subspace - Gram Schmidt Process - QR- Decomposition Best

### Research Article On Fuzzy Soft Linear Operators in Fuzzy Soft

tor spaces, the fuzzy soft Cauchy-Schwartz inequality, the fuzzy soft orthogonality and the fuzzy soft Hilbert spaces. Moreover, we deﬁne the fuzzy soft linear operators in fuzzy soft Hilbert spaces, establish their related theorems, intro-duce fuzzy soft spectral theory of them and prove the fuzzy soft self-duality of fuzzy soft Hilbert spaces.

### LECTURES FOR MLSP

o Inner product as projection o Outer product as modulation Interpreting matrix-on-vector o Transform of vector / transform of space Matrix operations perspectives o Transform between spaces o Four different views o Rules of operations The matrix as an object in a space Types of transformations o Identity o Stretching

### Curriculum & Syllabi of M.Sc. MATHEMATICS

theorem, invariant direct sums, Quotient space, Spectral theorem, Inner product spaces, Orthogonality, orthonormal sets, Bessel s inequality, Gram-Schmidt process , Unitary operators , Normal operators, Bilinear Forms, symmetric bilinear forms, Quadratic forms. L T P C 3 1 0 3

### Stability of Quadratic Functional Equation

equation on the whole inner product space. Pinsker  characterized orthogonally additive functionals on an inner product space when the orthogonality is the ordinary one in such spaces. Sundaresan  generalized this result to arbitrary Banach spaces equipped with the Birkhoff-James orthogonality.

### FUNDAMENTALS OF LINEAR ALGEBRA

FUNDAMENTALS OF LINEAR ALGEBRA James B. Carrell [email protected] (July, 2005)

### DOCTOR OF PHILOSOPHY IN MATHEMATICS

Metric spaces, normed spaces and Banach space linear operators, inner products and Hilbert spaces, Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, and closed graph theorem. 252516 Set-Valued Analysis 3(2-2-5) Limit of sets, continuity of set-valued functions, closed convex processes,

### MIT Art, Design and Technology University, Pune MIT School of

Vector spaces: definition, linear independence of vectors, basis, inner product and inner product space, orthogonality, Gram-Schmidt procedure, subspaces, Matrices: coordinate-dependent linear transformations, null and range spaces, Linear algebraic equations:

### SOME ALTERNATIVE CONCETS FOR FUZZY NORMED SPACE AND FUZZY 2

generalization of a fuzzy normed space to a fuzzy 2-normed space. Conversely, a fuzzy 2-normed space is reduced to a fuzzy normed space. The same method is applied to fuzzy inner product spaces. Keywords and phrases: fuzzy normed space, fuzzy 2-normed space, fuzzy point. Received July 2, 2015 References  J. Alonso and C. Benitez

### Programming in MATLAB

Programming in MATLAB 2.25 Gp.Capt.Thanapant Raicharoen, Ph.D. qWe can write the decision function for classifying thin and fat as follows: 21 21 1 (thin) if 1000

### INSTRUMENTATION AND CONTROL ENGINEERING

Linear Algebra and Matrices Vector spaces, Subspaces, Basis and Dimension - Systems of Linear Equations. Linear Transformations - Kernel and Image - Geometric Ideas - Inner Product spaces - Orthogonality - Orthonormal basis - Reflections and Orthogonal maps of the plane - Orthogonal complements and Projections.

### Linear Algebra Prentice Hall International Editions Taschenbuch

mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra.

### Orthogonality spaces allowing gradual transitions

from an inner-product space of the indicated type is then simply its linear dimension. We should remark that the central notion on which this work is based is actually quite unspeci c. We can hardly claim that De nition 2.1 suggests by itself an application in the context of quantum physical mod-elling. Indeed, orthogonality spaces are

### Lecture 4 Inner product spaces - University of Waterloo

The inner product in this space is given by hf,gi = Z ∞ −∞ f(x)g(x) dx, (14) Once again, L2 is the only Lp space for which an inner product exists. It is also a Hilbert space, and will the primary space in which we will be working for the remainder of this course. Orthogonality in inner product spaces

### Faculty School of Field of study Technomathematics (054)

1‐2 M1‐2 6 BM20A3101 Fuzzy Sets and Fuzzy Logic Attendance is required in both semesters 2‐3 M1‐2 6 BM20A5600 Inverse Problems and Sparse Transforms 1 M1‐2 3 BM20A2500 Linear Algebra and Normed Spaces 1‐4 M 1 4 BM20A2201 Logic and Discrete Methods

### Linear Logic for Generalized Quantum Mechanics

Quantum mechanics is based on a Hilbert space of states, a metrically complete inner product space, made an orthoframe by interpreting orthogonality standardly for Euclidean space. Although completeness does not have a rst-order de nition for either inner product spaces or orthoframes [Gol84], the condition of orthomodularity

### Soft Sets - Motivation and Overview

Samantha presented an idea of soft inner product on soft linear spaces and some of their properties are investigated. Soft Hilbert spaces, orthogonality and orthonormality in soft Hilbert spaces are also studied. In 2014 Das Samantha introduced Soft vectors in soft linear spaces and their properties are investigated.

### Fuzzy Farthest Points and Fuzzy Best Approximation Points in

fuzzy inner product space in linear space. Since than some some math-ematicians have de ned fuzzy metrics and norms on a linear space from various points of view [6,9, 27, 29]. In 1994, Cheng and Mordeson intro-duced a de nition of fuzzy norm on a linear space in such a manner that the corresponding induced fuzzy metric is of Kramosil and

### www.oyama-ct.ac.jp

Fuzzy inner product spaces and fuzzy orthogonality ,Advances in Fuzzy Sets and Systems, , 1-14, (2010.4) 30, 186-189 , (2010.8) super Frobenius formula for the