I have been studying bra-ket notation for QM and have come across the concept of dual spaces and their relation to bra vectors. I can appreciate that the definition of a dual space is such that, given a vector space $\mathcal{V}$ on a field $\mathbb{F}$ the dual space is the set of all linear functionals $\phi: \mathcal{V} \rightarrow \mathbb{F}$. I also accept that for finite dimensional vector spaces there is an isomorphism between the space and its dual (and that for infinite dimensional spaces one can use the Riesz-Representation theorem guaranteeing a correspondence to the topological dual of $\mathcal{V}$).
Clearly, however I don't appreciate these topics enough to be able to justify why any of the ideas are required in QM. Why, for example, does the dual space (or topological dual space) even need to be considered for QM? Why does there have to be a 1-1 correspondence between bra and ket vectors for the formalism to work? Surely some alternate mathematical framework could be used just on the vector space itself (e.g. $\langle a|b \rangle$ might just be defined as being an operation between two ket vectors $\in \mathcal{V}$) where the idea of a dual space isn't needed?
Edit
Using the definition here where for a ket vector $|\phi \rangle$ we define the functional $f_{\phi}= \langle\phi|$ by $(|\phi \rangle, |\psi \rangle)$ where $( , )$ denotes the inner product on the Hilbert space, what is the use of considering the functionals as a dual space? Surely, all the useful information is contained in the above definition and only the ket vector space is needed?