I need someone to corroborate this argument.
I have a complex pre-Hilbert space $(V,<,>_{V})$ with an action denoted by $\cdot_{V}$.
When I complet $V$ in order to have a Hilbert space $(H,<,>_{H})$, but the actioon of scalars over $H$ have to be extended too, let's call it $\cdot_{H}$.
Is it fine to define the multiplication by $i$ in the following way? given $h\in H$, I can consider a sequence $\{h_{n}\}_{n\in\mathbb{N}}\subseteq V$ such that $h_{n}\to h$ in $H$, and then say: $i\cdot_{H} h:=lim_{n} i\cdot_{V} h_{n}$.
The limite on the RHS must exist because $i\cdot_{V} h_{n}$ is a Banach sequance in $V$, then it must have a limite in $H$.
Is this the way for complete the action?