Bessel sequences, frames, tight frames,independent frames and Riesz basis of order p for a Banach space X are introduced and discussed. It is proved that the set BpX Bessel sequences of order p in X is a Banach space; it is established that a sequence f = {fn}n∈Λ in X is a Bessel sequence of order p if and only if c>0 such that ‖∑n∈Λcnfn‖≤c‖{cn}‖q for all {cn}∈lq, where p-1+q-1=1; it is also proved that the spaces BpX and B(X*, lp) are linearly and isometrically isomorphic. In light of operator theory, it is shown that a Bessel sequence f in X is a frame of order p for X if and only if the operator Tf is bounded below, and it is a Riesz basis of order p for X if and only if Tf is invertible. Moreover, it is proved that independent frames of order p and Riesz bases of order p for X are the same. Lastly, it is shown that a Banach space X has a Riesz basis of order p if and only if there exists a norm ‖·‖0 equivalent to the original norm of X such that (X，‖·‖0) is isometrically isomorphic to lq.

Banach space; Bessel sequence of order p; frame of order p; Riesz basis of order p