on the dual of a left quaternionic Hilbert space.)

Separable quaternionic Hilbert spaces admit countable orthonormal bases. Let $V_H^L$ be a left quaternionic Hilbert space and let $\{e_v\}_{\nu = 0}^N$ ($N$ could be finite or infinite) be an orthonormal basis for it. Then, $\langle e_\nu \mid e_\mu\rangle = \delta_{\nu \mu}$ and any vector $f \in V_H^L$ has the expansion $f = \sum_\nu f_\nu e_\nu$, with $f_\nu = \langle f\mid e_\nu\rangle \in H$. Using such a basis, it is possible to introduce a multiplication from the right on $V_H^L$ by elements of $H$. Indeed, for $f \in V_H^L$ and ${\mathbf{q}}\in H$ we define, $$f{\mathbf{q}}= \sum_\nu (f_\nu {\mathbf{q}})e_\nu .
\label{rightmult}$$

The field of quaternions $H$ itself can be turned into a left quaternionic Hilbert space by defining the inner product $\langle {\mathbf{q}}\mid {\mathbf{q}}^\prime \rangle = {\mathbf{q}}{\mathbf{q}}^{\prime\dag} = {\mathbf{q}}\overline{{\mathbf{q}}^\prime}$ or into a right quaternionic Hilbert space with $\langle {\mathbf{q}}\mid {\mathbf{q}}^\prime \rangle = {\mathbf{q}}^\dag {\mathbf{q}}^\prime = \overline{{\mathbf{q}}}{\mathbf{q}}^\prime$.

### Quaternionic Hilbert Spaces of Square Integrable Functions

Let $(X, \mu)$ be a measure space and $H$ the field of quaternions, then $$L^2_H(X,\mu)=\left\{f:X\rightarrow H\;\; \left| \;\; \int_X|f(x)|^2d\mu(x)<\infty \right.\right\}$$\[L\^2\] is a left quaternionic Hilbert space, with the (left) scalar product $$\langle f \mid g\rangle =\int_X f(x)\overline{g(x)} d\mu(x),
\label{left-sc-prod}$$ 