Extension of frames and bases - I

We extend the theory of operator-valued frames (resp. bases), hence the theory of frames (resp. bases), for Hilbert spaces and Hilbert C*-modules, in two folds. This extension leads us to develop the theory of operator-valued frames (resp. bases) for Banach spaces. We give a characterization for the operator-valued frames indexed by a group-like unitary system. This answers an open question asked in the paper titled "Operator-valued frames" by Kaftal, Larson, and Zhang in \textit{Trans. Amer. Math. Soc.} (2009). We study stability of the extension. We also extend Riesz-Fischer theorem, Bessel's inequality, variation formula, dimension formula, and trace formula. Further, notions of p-orthogonality, p-orthonormality and Riesz p-bases have been developed in Banach spaces and Paley-Wiener theorem has also been generalized. We derive `4-inequality,' `4-parallelogram law,' and `4-projection theorem.'