This is an extract from my third year 'Foundations of QM' lecture notes: If ψ1 and ψ2 are admissible states, then the superposed state [itex]\alpha[/itex]ψ1 + ψ2[itex]\beta[/itex] ( [itex]\alpha[/itex],[itex]\beta[/itex] [itex]\in[/itex] C ) is also an admissible state. [itex]\rightarrow[/itex] complex vector space. I understand that a linear superposition of allowed state is also an allowed state. It follows from the linearity of the Schrodinger's equation. What I fail to understand is how this leads to the concept of a complex vector space in QM. Any help would be greatly appreciated.