On Structral Properties of Some Banach Space-Valued Schröder Sequence Spaces

Some properties on Banach spaces, such as the Radon–Riesz, Dunford–Pettis and approximation properties, allow us to better understand the naive details about the structure of space and the robust inhomogeneities and symmetries in space. In this work we try to examine such properties of vector-valued Schröder sequence spaces. Further, we show that these sequence spaces have a kind of Schauder basis. We also prove that ℓ1S,V possesses the Dunford–Pettis property and demonstrate that ℓpS,V satisfies the approximation property for 1≤p<∞ under certain conditions and ℓ∞S,V has the Hahn–Banach extension property. Finally, we show that ℓ2S,V has the Radon–Riesz property whenever V has it.