where $\overline{g(x)}$ is the quaternionic conjugate of $g(x)$, and (left) scalar multiplication $af, \; a\in H,$ with $(af)(q) = af(q)$ (see [@Vis] for details). Similarly, one could define a right quaternionic Hilbert space of square integrable functions.

Dual spaces
-----------

In order to obtain a quaternionic version of the Riesz representation theorem, we need to recall a few facts about the dual space of a quaternionic Hilbert space. We follow [@Alek] in order to do this. Let $\mathfrak{H}_{ld}$ be the left dual space of the left quaternionic Hilbert space $V_H^L$. That is, $$\mathfrak{H}_{ld}=\{h : V_H^L\longrightarrow H~ \mid ~h~ \text{is bounded and left linear}\}$$ with the usual norm $\|h\|= \sup\{\vert h(x)\vert \;\mid \;\|x\|=1,~x\in V_H^L\}$. It is known that $\mathfrak{H}_{ld}$ is a real vector space. Moreover, $\mathfrak{H}_{ld}$ can be transformed into a quaternionic Hilbert space. Indeed, as noted above, $V_H^L$ also admits a right multiplication by quaternionic scalars. Using this fact, for any functional $h\in\mathfrak{H}_{ld}$ and any $\lambda\in H$ define $$(\lambda h)(x)=h(x\lambda),\quad (h\lambda)(x)=h(x)\lambda;~~~~~x\in V_H^L.$$ Then $\mathfrak{H}_{ld}$ becomes a two-sided quaternionic Banach space, with the scalar multiplication so defined.

For any functional $h\in\mathfrak{H}_{ld}$, exactly as in the real and complex cases, $$\label{E13}
h(x)=\langle x\vert y\rangle,\quad x\in V_H^L$$ for a vector $y\in V_H^L$, and 