# Norm derivatives and characterizations of inner product spaces

• 188 Pages
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World Scientific , Singapore, Hackensack, NJ
Normed linear spaces, Inner product s
Classifications The Physical Object Statement Claudi Alsina, Justyna Sikorska and M. Santos Tomas Contributions Sikorska, Justyna, Tomás, M. Santos (Maria Santos) LC Classifications QA322.4 .A47 2010 Pagination x, 188 p. : Open Library OL25342873M ISBN 10 9814287261 ISBN 13 9789814287265 LC Control Number 2010277952 OCLC/WorldCa 540206466

The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed : Claudi Alsina, Justyna Sikorska, M Santos Tomas.

This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces.

Norm Derivatives and Characterizations of Inner Product Spaces The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces.

This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces.

Characterizations of Inner Product Spaces. Authors: Amir. Free Preview. Buy this book eB49 Norm Derivatives Characterizations.

*immediately available upon purchase as print book shipments may be delayed due to Norm derivatives and characterizations of inner product spaces book COVID crisis. ebook access is temporary and does not include ownership of the ebook. Only valid for books with an.

Characterizations of Inner Product Spaces. Authors (view affiliations) Dan Amir; Book. 95 Citations; Norm Derivatives Characterizations. Dan Amir. Pages James’ Isoceles Orthogonality (Midpoints of Chords) Dan Amir. Pages About this book.

Keywords. The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical. SOME CHARACTERIZATIONS OF INNER-PRODUCT SPACES BY MAHLON M. DAY 1.

### Details Norm derivatives and characterizations of inner product spaces FB2

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 defined from an inner product. It should be emphasized that these are not. The standard inner product between matrices is hX;Yi= Tr(XTY) = X i X j X ijY ij where X;Y 2Rm n.

Notation: Here, Rm nis the space of real m nmatrices. Tr(Z) is the trace of a real square matrix Z, i.e., Tr(Z) = P i Z ii. Note: The matrix inner product is the same as our original inner product File Size: KB.

I understand that there can be many different types of norms (e.g. mean norm, Cartesian norm, supremum norm etc). Are there also other types of inner products apart from $\langle x,y \rangle= \sum_{j =1}^n x_j y_j$?Also, I read that for any inner product on a vector space V the function $\|x\| = \sqrt{\langle x,x \rangle}$ defines a norm on the vector space.

Norm Derivatives and Characterizations of Inner Product Spaces Volterra, Fredholm and Hilbert, and made it possible to prove strong re- sults, such as the Hahn-Banach or Banach-Steinhaus theorems. Abstract Hilbert spaces were introduced by von Neumann in in an axiomatic way, and work on abstract normed linear spaces was done by Wiener, Hahn and Helly.

CHARACTERIZATIONS OF INNER PRODUCT SPACES BY STRONGLY CONVEX FUNCTIONS A rich collection of such characterizations is con- Alsina, J. Sikorska and M.S. Tom´as, Norm derivatives and characterizations of inner product spaces, Hackensack, NJ: World Scientiﬁc, 5. Amir, Characterizations of inner product spaces, Operator Theory.

Norm Derivatives and Characterizations of Inner Product Spaces Dec 2, by Claudi Alsina, Justyna Sikorska, M Santos Tomas.

If φ′+(x,y) = φ′-(x,y) for all y ∈ E then the common value φ′ x (y) is a linear functional on y ∈ E (called: the derivative φ′ x of φ at x) This is a preview of subscription content, log in to check : Dan Amir. Norm derivatives: Definition and basic properties.

Orthogonality relations based on norm derivatives. ρ′ ±-orthogonal transformations. On the equivalence of two norm derivatives. Norm derivatives and projections in normed linear spaces. Norm derivatives and Lagrange's identity in normed linear spaces.

On some extensions of the norm derivatives. The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces.

Since the appea. INNER PRODUCTS We denote by dim(X) the dimension of X, and we write X is (H) when the norm of X derives from an inner product; in this case, the inner product (x, y) is given by r(x, y) \\x\\.

PRESERVING ANGLES OR ORTHOGONALITY CHARACTERIZATIONS We state some well-known results that we shall use m the by: 6. Offers an overview of the characterizations of real Norm ed spaces as inner product spaces based on Norm derivatives and generalizations of the most basic geometrical properties of triangles in Norm ed spaces.

This book presents various methods of characterizing inner. Inner Product Spaces. In making the deﬁnition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products.

"This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives.

It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces.

1 Orthogonal Basis for Inner Product Space If V = P3 with the inner product = R1 −1 f(x)g(x)dx, apply the Gram-Schmidt algorithm to obtain an orthogonal basis from B = {1,x,x2,x3}.

2 Inner-Product Function Space Consider the vector space C[0,1] of all continuously diﬀerentiable functions deﬁned on the closed interval [0,1].File Size: KB. INNER PRODUCT AND NORM 11 Deﬁnition An inner product on a real vector space V is a real function hx,yi: V×V→R such that for all x, y, z in V and all c in R, (1) hx,yi = hy,xi (2) hcx,yi = chx,yi (3) hx+y,zi = hx,zi+hy,zi (4) hx,xi > 0 if and only if x 6=0.

An inner product on a complex vector space is de ﬁned similarly. The inner. An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space.

A complete space with an inner product is called a Hilbert space. An (incomplete) space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm. Norm inequalities and characterizations of inner product spaces Author: A.

Amini-Harandi, M. Rahimi and M. Rezaie Subject: Math. Inequal. Appl., 21, 1 () Keywords: 46C15, 46B20, inner product space, characterizations of inner product spaces, Dunkl-Williams inequality Created Date: 1/1/ PM. Rätz, Characterizations of inner product spaces by means of orthogonally additive mappings, Aequationes Math.

58 (), – Mathematical Reviews (MathSciNet): MR Zentralblatt MATH: By Remark 1 and Theorem 1, follows each real 2-normed space with strictly convex norm by module c >0 is strictly convex.

3 Characterization of 2-inner product By Example 2 we get, 2-norm (7) is strictly convex of module 1. On the other hand, in [4] is proved that (,) (,) (, |) (,) (,) x yxz xy z zy zz = defines 2-inner product, in which the. For each vector u 2 V, the norm (also called the length) of u is deﬂned as the number kuk:= p hu;ui: If kuk = 1, we call u a unit vector and u is said to be normalized.

For any nonzero vector v 2 V, we have the unit vector v^ = 1 kvk v: This process is called normalizing v. Let B = u1;u2;;un be a basis of an n-dimensional inner product space vectors u;v 2 V, write. SOME CHARACTERIZATIONS OF INNER PRODUCT SPACES BY INEQUALITIES DAN S¸TEFANMARINESCU,MIHAI MONEA ANDMARIANSTROE Abstract.

In this paper we present some new characterizations of inner product spaces by us-ing inequalities. First, we explore a classical idea consisting in the transformation of the par-allelogram law into an inequality. As the main tool, we use a fixed point theorem for the function spaces.

We finish the paper with some new inequalities characterizing the inner product spaces. Introduction. In the literature there are many characterizations of inner product spaces. The first norm characterization of inner product space was given by Fréchet in Cited by: Browse other questions tagged real-analysis functional-analysis inner-product-space derivatives or ask your own question.

The Overflow Blog Introducing Collections on Stack Overflow for Teams. In mathematics, a normed vector space is a vector space on which a norm is defined.

### Description Norm derivatives and characterizations of inner product spaces PDF

A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that has the following properties. The zero vector, 0, has zero length; every other vector has a positive length.The purpose of this book is to give systematic and comprehensive presentation of theory of n-metric spaces, linear n-normed spaces and n-inner product spaces (and so 2-metric spaces, linear 2-normed spaces and 2-linner product spaces n=2).

Since andS. Gahler published two papers entitled "2-metrische Raume und ihr topologische Strukhur" and "Lineare 2-normierte Raume", a .1 A characterisation of inner product spaces in terms of the maximal circumradius of spheres Deﬁnition (Inner product space, non inner product space).

Let (X,kk) be a normed real vector space. We call (X,kk) an inner product space, or short ips, if there exists an inner product h,i on X, which induces the norm, i.e. kxk = pAuthor: Sebastian Scholtes.