then $\|h\|=\|y\|$.

Let $\{e_{\nu}\mid \nu\in\Lambda\}$ be a fixed orthonormal basis of $V_H^L$ and define $J:\mathfrak{H}_{ld}\longrightarrow V_H^L$ by the cannonical mapping defined by the relation (\[E13\]), that is by, $h(x)=\langle x\vert Jh\rangle;~~x\in V_H^L$. Then it can be shown that $J$ is additive, isometric and bijective. Now define $$K:V_H^L\longrightarrow V_H^L, \qquad K(x)=\sum_{\nu\in\Lambda}\overline{\langle x\vert e_{\nu}\rangle}e_{\nu}.$$ Clearly, $K$ is additive. (Note, $x = \sum_{\nu\in\Lambda}\langle x\vert e_{\nu}\rangle e_{\nu}$).

The left dual space $\mathfrak{H}_{ld}$ of $V_H^L$, is also a two-sided quaternionic Hilbert space, if we introduce the inner product in $\mathfrak{H}_{ld}$ by $$\label{E14}
\langle h\vert k\rangle=\langle KJh\vert KJk\rangle;~~~~~h,k\in\mathfrak{H}_{ld}.$$ The inner product (\[E14\]) is consistent with the norm of $\mathfrak{H}_{ld}$.

Further, if we define $U:\mathfrak{H}_{ld}\longrightarrow V_H^L$ by $U=KJ$, then $U$ is a two-linear bijective map, and thereby $\mathfrak{H}_{ld}$ is isomorphic to $V_H^L$.

Cullen regular functions
------------------------

There have been a number of different suggestions in the literature on how the notion of holomorphy could be extended to functions of a quaternionic variable. We mention two here, of which we shall adopt the second definition. For a brief history of quaternionic holomorphy we refer the reader to [@Am]. The first definition of quaternionic holomorpy is given via the Cauchy-Fueter equations, which attempts to mimic the Cauchy-Riemann equations in 