الفهرس 
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Abstract
A topological vector space X is, roughly speaking, a set which carries
two structures: a structure of topological space; a structure of vector space.
Furthermore, some kind of compatibility condition must relates these
two structures on X. one of the most important classes of topological vector
spaces is the class of locally convex spaces. In fact, the theory of locally
convex spaces is significantly richer in results than the theory of topological
vector spaces, chiefly because there are always plenty of continuous linear
functionals on an locally convex space. Moreover, almost all the concrete
spaces that occur in functional analysis are locally convex. In a sense, semi-
-norms and duality play a vital role in the theory of locally convex spaces:
In fact, to introduce a topology in a linear space of infinite dimension suitable
for applications to classical and modem analysis, it is sometimes necessary to
make use of a system of an infinite number of semi-norms. If the system
reduces to a single semi-norm, the corresponding linear space is called a
normed space. If further more, the space is complete with respect to the
topology defined by this semi-norm, it is called a Banach space. Also, duality
is what makes this theory powerful because it establishes a tool to translate a
problem on the space (where it may appear to be difficult) into one
concerning its linear forms (which may happen to be much easier to handle).
Duality also admits the replacement of the original topology by simpler ones
when dealing with problems involving boundedness convexity, continuity,
etc. One of the most important of these topologies is the weak topology on a
given locally convex space.
This thesis is devoted to study some cocepts in locally convex spaces,
viz., weakly boundedness, duality, some linear topologies (J3-topologies,
Mackey topologies, ’[ pc topologies , etc.), equicontinuous sets and
compactologies, bilinear forms and topological tensor products. In a trial to generalize the n-kolmogorov diameters, 60 (B), of bounded
subsets in normed spaces, we arrive to introduce the notions of outer and
mner s-diameters, dn (B,E) and 6n (B,E) , of a bounded subset B. The notion
of quasi-compactness of a given subset B is also given. Many results we have
obtained in this direction are collected in our first paper which have
submitted” for publication under the title ”Outer and Inner E - Diameters of
Bounded Sets.
The notions of outer and inner E- diameters helped us to introduce the
notions of diametral numbers of a bounded linear operator. All results we
have obtained in this direction are also collected in our second paper which
is also submitted for publication under the title.
”Diameteral Numbers of Bunded Linear Operators in Normed Space. II
This thesis consists of five chapters
Chapter (0) :
This chapter contains the basic definitions and notations which we are
needed in the thesis.
Chapter 1 :
This chapter is devoted to the study of the theory of locally convex
spaces and how we can generate a locally convex space by means of an
infinite system of semi-norms. The study of the duality theory for locally
convex spaces, and some linear topologies on functional spaces and
equicontinuity are our main interests in this chapter.
Chapter 2:
Our main intersets of study are the properties of the projective topology ’trc
defined on the tensors product X ® Y of two locally convex spaces X and Y
as the strongest locally convex topology on X ® Y such that the canonical
bilinear map ® : X x Y -e-X ® y :(x,y) --)0 x ® y, is continuous. A short survey on the algebraic theory of tensor products and the injective
norm E are also contained in this chapter.
Chapter (3) :
In this chapter we introduce the notions of outer and inner s-diameters,
fln (B,E) and 8n (B,E), of a bounded subset B in a normed space X.
Also we introduce the notion of quasi-compactness of a bounded subset B in
a normed space X. Finally we obtain some properties of the outer and inner
entropy numbers, en (B) and fn (B), of a bounded set B.
Chapter 4 :
This chapter is devoted to the study of some important classes of
compact and nuclear linear operators. We make use of the above notions of
outer s-diameters fln (B,E) and inner s-diameters 8n (B,E) to introduce the
notions of the diameteral numbers, E-Dn (T) and E-dn (T), of a bounded linear
operator T. Also we give a modified definitions of a compact linear operator
by means of its diametral numbers E-Dn (T).