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In this paper, we will generelize $b$-frames; a new concept of frames forHilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banachspace and see how it can be a frame for a Hilbert space. Instead of the scalarproduct we will use a new product called the $b$-dual product and it isconstructed via a bilinear mapping. We will introduce new results about thisproduct, about $b$-frames, and about $K$-$b$-frames, and we will also give someexamples of both $b$-frames and $K$-$b$-frames that have never been givenbefore. We will give the expression of the reconstruction formula of theelements of the Hilbert space. We will as well study the stability andpreservation of both $b$-frames and $K$-$b$-frames; and to do so, we will givethe equivalent of the adjoint operator according to the $b$-dual product.

$K$-$b$-frames for Hilbert spaces and the $b$-adjoint operator