الفهرس 
IntroductionOnly 14 pages are availabe for public view
 |  | from | 94 |  |  |
| | | from | 94 | | |
Abstract
It is well known that a normed space X is uniformly convex (smooth)
if and only if its dual X∗ is uniformly smooth (convex). We extend
some of these geometric properties to the so-called 2-normed spaces
and also we introduce definition of uniformly smooth 2-normed spaces.
We get some fundamental links between Lindenstrauss duality formulas.
Besides, a duality property between uniform convexity and uniform
smoothness of 2-normed space is also given. Moreover, we introduced
a definition of Metric projection in a 2-normed space and also theorem
of the approximation of fixed points in 2-Hilbert spaces.
1. In chapter 1: we give a summary of quotient spaces and state
the Hahn-Banach Theorem; which is the one of the fundamental
theorems of functional analysis. We study some of the geometric
properties in linear normed spaces and we give a brief history of
Metric projection existence and uniqueness in different spaces.
2. In chapter 2: we discuss some spaces like 2-metric spaces and
linear 2-normed spaces. We define a Cauchy sequence and a convergent
sequence in both spaces and mention the relation between
both concepts and also the notions of bounded bilinear function.
3. In chapter 3: we study the completion of linear 2-normed spaces.
4. In chapter 4: we study some geometric properties for linear 2-
normed spaces like strictly convex, strictly 2-convex, uniformly
convex and uniformly 2-convex 2-Banach spaces. We give new
definitions and prove new results. We prove an important result
which state that: “a 2-normed space X is uniformly convex if
and only if all dual spaces (X/V (c))∗ or X∗
c , for all c ̸= 0 in X
are uniformly smooth”. It is our purpose to extend the notion of
4
CONTENTS 5
uniform smoothness for Banach spaces to 2-normed spaces. Moreover,
by extending some theorems for uniformly convex uniformly
smooth Banach spaces to the case of 2-normed spaces, we prove
the existence of the metric projection point in uniformly convex
2-normed spaces and also the fixed point theorem of non-self maps
in 2-Hilbert spaces.