In a set of notes it is stated that: Given *coherent states* of a harmonic oscillator $$| \alpha \rangle = \pi^{-\frac{1}{2}} \text{exp}(-\frac{1}{2}|\alpha|^2)\sum\_{n = 0}^{\infty} \frac{\alpha^n}{(n!)^{\frac{1}{2}}}|n \rangle~~~~~~~~~~(1)$$where $|n \rangle$ is the nth energy eigenstate of the oscillator. We can define the completeness relation as the integral $$\int d^2 \alpha | \alpha \rangle \langle \alpha | = \sum\_{n=0}^{\infty}| n \rangle \langle n| = 1~~~~~~~~~~~~~~~~(2)$$ But these states are not orthonormal. They do not span the Hilbert space in the two-dimensional $\alpha$ space.

**Question:** How is the "two dimensional $\alpha$ plane" defined? If $\{ | \alpha \rangle \}$ satisfies completeness relation (2) then what space does it span?