Banach spaces are a different generalization of Hilbert spaces. In a Banach space , the vectors may be notated by kets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.
banach spaces are a different generalization of hilbert spaces in a banach space  the vectors may be notated by kets and the continuous linear functionals by bras over any vector space without topology we may also notate the vectors by kets and the linear functionals by bras in these more general contexts the bracket does not have the meaning of an inner product because the riesz representation theorem does not apply