| In this thesis, Firstly, the space Ab(X) of approximate bounded sequence over the Frechet space X is researched, the space Ab(X) of approximate bounded sequence is a non-separable Frechet space is proved.Especially,when X is a Banach space,the spaceAb(X) is not locally convex and locally bounded;When X is the real or complex scalar field K, the duality theorem of the spaceAb(K) is partly resolved. Secondly, based on the concept of the approximate bounded sequence space, the concepts of absolutely average bounded sequence space Aab(X) and average bounded sequence space Avb(X) are defined over Banach space X, the equivalent conditions of two bounded property are obtained,and the space Aab(X) and the space Avb(X) are non-separable,non-reflexive Banach space which have not the Krefn-Mil'man property and Radon-Nikodym property are proved.lastly,the mutual relations of the bounded sequence spaces over Banach space X are discussed. |
