In this paper, we prove the following results. There exists a Banach space
without basis which has a Schauder frame. There exists an universal Banach
space B (resp. B~) with a basis (resp. an unconditional basis) such
that, a Banach X has a Schauder frame (resp. an unconditional Schauder frame
) if and only if X is isomorphic to a complemented subspace of B (resp.
B~). For a weakly sequentially complete Banach space, a Schauder frame
is unconditional if and only if it is besselian. A separable Banach space X
has a Schauder frame if and only if it has the bounded approximation property.
Consequenty, The Banach space L(H,H) of all
bounded linear operators on a Hilbert space H has no Schauder
frame. Also, if X and Y are Banach spaces with Schauder frames then, the
Banach space XββΟβY (the projective tensor product of X
and Y) has a Schauder frame. From the FaberβSchauder system we construct a
Schauder frame for the Banach space C[0,1] (the Banach space of continuous
functions on the closed interval [0,1]) which is not a Schauder basis of
C[0,1]. Finally, we give a positive answer to some open problems related to
the Schauder bases (In the Schauder frames setting)