Grothendieck özelliğine sahip Banach uzayları

Let X be a Banach space and X-star be its dual space. On X-star there are three standardtopologies, namely norm topology, weak topology and weak-star topology. Thesetopologies are distinct. For certain Banach spaces X, including several familiar ones,weak-star convergent sequences in X-star converges weakly. These spaces are said tohave the Grothendieck property. A Banach space possessing this property is said to be theBanach space with the Grothendieck property or simply a Grothendieck space. The trivialexamples of such spaces are the reflexive Banach spaces. The first nontrivial one, thespace of bounded sequences, is presented by A. Grothendieck in 1953. One can easilyfind weakly compact operators from a Grothendieck space. In his paper ?Sur lesapplications lineaires faiblement compactes d'espaces du type C(K),? Grothendieckproved a criterion for weak compactness in the dual of C(K), the space of continuouscomplex valued functions on K, where K is any compact Hausdorff space. A couple ofyears later A. Pelczynski gave a measure theoretic proof of Grothendieck's criterion andintroduced the notion that is named after him as Pelczynski's property V. We say that aBanach space has the Pelczynski's property V if any unconditionally converging operatoris weakly compact. Any dual space having the Pelczynski's property V is a Grothendieckspace. In particular, any von-Neumann algebra is a Grothendieck space. The mainobjective of this Ms. thesis is to learn the basics of the Banach space theory, to study theBanach spaces with the Grothendieck property and make a synthesis of the resultsobtained so far.