  In this paper, we will generelize $b$-frames; a new concept of frames for
Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach
space and see how it can be a frame for a Hilbert space. Instead of the scalar
product we will use a new product called the $b$-dual product and it is
constructed via a bilinear mapping. We will introduce new results about this
product, about $b$-frames, and about $K$-$b$-frames, and we will also give some
examples of both $b$-frames and $K$-$b$-frames that have never been given
before. We will give the expression of the reconstruction formula of the
elements of the Hilbert space. We will as well study the stability and
preservation of both $b$-frames and $K$-$b$-frames; and to do so, we will give
the equivalent of the adjoint operator according to the $b$-dual product.
