We study new sequence spaces associated to sequences in normed spaces and the band matrix F̂ defined by the Fibonacci sequence. We give some characterizations of continuous linear operators and weakly unconditionally Cauchy series by means of completeness of the new sequence spaces. Also, we characterize the barreledness of a normed space via weakly∗ unconditionally Cauchy series in [Formula: see text].
Keywords: Fibonacci sequence spaces; barreledness; completeness; weakly unconditionally Cauchy series.