الفهرس 
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Abstract
Functional analysis has risen of the beginning of the twentieth century. Functional analysis is concerned with sequences or functions which are studied in vector spaces. In this thesis we describe some properties concerning sequence spaces, i.e., those Banach spaces which can be presented in some natural manner as spaces of sequences. The current thesis deals with two of the important properties in functional analysis. It is an extension of the work of many fine mathematicians. It focuses on two important and central questions, which could be phrased as follows: Does a Banach space have the approximation property? The other, Does a Banach space have the property H? We have attempted that questions our attempts were satisfactory and
acceptable to answer the first question we may follow the following direction the origin of the approximation property was The following fundamental result in the theory of operators on Hilbert spaces ,and which asserts that the compact operators on Hilbert space are operators which are limits in norm of operators of finite rank, one part of this assertion, namely that, every TE B(X, Y) for which liT - Tnll ~O for sequence {Tn}~=l EB (X , Y) with dim TnX < 00 is compact. This result is true for every pair of Banach spaces X and Y. The converse assertion is true for many examples spaces of X and Y
besides Hilbert space the question whether the converse assertion is true for or arbitrary Banach spaces X and Y which was called the approximation problem. Our purpose we study a Banach space which one of many variants of the approximation property, i. e., we study a Banach space for which every compact TE B(X , Y ) or TE B(Y ,X ) is limit in norm of sequences of . the finite rank operators. Every Banach space satisfied the approximation property. Grothendieck was unable to prove equivalent formulation of the approximation property. Enflo solved this problem, but his solution a negative solution to the basic problem.