I am a physicist in training, and was only taught about the dual vector space quite loosely within the context of quantum mechanics. I was told that the dual vector space to some ket space in which the kets are column vectors, consists of row vectors with elements from the same field.
Now I am reading the formal definition for a dual space as a "space of all linear functionals $f:V\rightarrow \mathbb{F}$".
Now I am happy with the idea that this itself forms a linear vector space. However I have not been able to convince myself- conceptually nor with a formal proof- that the dual space to a linear vector space consisting of some n dimensional column vectors is necessarily isomorphic to a linear vector space of row vectors.
If my understanding is correct, the loose notion of a dual space I was introduced to via QM is actually saying that "a linear functional $f:V\rightarrow \mathbb{F}$ acting on a linear vector space $V$ in which the elements are some n component column vectors with entries from a field $\mathbb{F}$, say {$v_i$} as the components of a vector, necessarily takes the form $\Sigma a_i v_i$ for any constants $a_i \in \mathbb{F}$. In this case there is clearly an isomorphism between the space of row vectors plus standard inner product (acting on the elemtns of the row vector space $V$) and the space of functionals on V.
However it is not clear to me that all linear functionals on the row vector space $V$ can be written in this form, and therefore that this does constitute the dual space to V.