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Recently, the authors interested some new problems on multiplier spaces of Lorentz’ almost convergence and  f  λ -convergence as a generalization of almost convergence.  f  λ -convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed space  X  and its continuous dual  X  * . In the present paper, we deal with  f  λ -convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces.

On some classical properties of normed spaces via generalized vector valued almost convergence