This is defined just like any other tensor product of two vector spaces (which is the Cartesian product, equipped with an intuitive definition of addition and multiplication). Since the space of bras is a vector space, it can be tensored with another vector space such as the space of kets. They also form a vector space, and they exist even if we don't define an inner product on the set of kets. "Bra"s are covectors, aka one-forms, defined as linear functions from a vector space to its field of scalars. a set of objects on which vector-vector addition and vector-scalar multiplication is defined (for some field of scalars). "Kets" are vectors in a vector space, i.e. The confusion is coming from the fact that you're thinking in terms of the bra-ket physics notation without understanding how the underlying vector spaces are constructed.
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