Problems with this alternative definition of dual space in QM? This question came up during discussions with my friends in QM, and they asked me this question I couldn't answer.
Mathematically, the dual vector space $V^*$ of a vector space $V$ is defined as the set of linear functions $f:V\to F$.
Let us consider only Hilbert spaces now. Then, by the Riesz Representation Theorem, an equivalent description of $V^*$ becomes

$u^*$ is a dual vector iff there exists a corresponding vector $u$ such that for any vector $v\in V$, $u^*(v)$ = $\langle u,v\rangle$.

Now, my friends asked this question - what if we defined dual vectors by the above statement? Would we come across any problems, either in QM or Math in general?
 A: As a quick notational point, given some vector space $V$ over a field $\mathbb F$, $V^*$ typically refers to the algebraic dual space, i.e. the vector space of linear maps from $V\rightarrow \mathbb F$.  If $V$ is in particular a topological vector space (so it has some notion of continuity, convergence, etc), then we can define the topological dual space $V'$ to be the vector space of continuous linear maps.  Note that $V^*=V'$ iff the vector space is finite-dimensional; otherwise $V^* \supsetneq V'$.
The Riesz representation theorem says that a Hilbert space $\mathcal H$ is isomorphic to its topological dual $\mathcal H'$.  From that point of view, there would be no problem in defining the topological dual space of a Hilbert space $\mathcal H$ to be the set of maps from $\mathcal H\rightarrow \mathbb C$ of the form $\langle \psi,\bullet\rangle$ for some $\psi\in \mathcal H$.  From this definition, continuity and linearity follow straightforwardly.
However, the notion of a topological dual requires only a topology, and does not need the full power of an inner product.  As an example, any normed vector space $B$ has a norm-induced topology and therefore a topological dual, but without an inner product structure, your proposed definition of $B'$ would not make sense.  And of course, topologies can exist even on non-normed vector spaces.
Also, I've assumed that you were referring to the topological dual space rather than the algebraic one. If you meant to ask whether it would be problematic to define $V^*$ in this way, then the answer is unambiguously yes; for any infinite dimensional Hilbert space, there are elements of $V^*$ which cannot be written in the form $\langle \psi,\bullet\rangle$ for any $\psi\in \mathcal H$.
