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# Math 401, Fall 2025: Thesis notes, S1, Complex projective space.
> [!CAUTION]
>
> In this section, without explicitly stated, all dimensions are in the complex field.
A complex projective space is a space that is the set of all lines through the origin in a complex vector space.
Described by that nature, there exists a natural definition of the complex projective space given as follows:
$$
\mathbb{C}P^n=\frac{\mathbb{C}^{n+1}\setminus\{0\}}{\sim}
$$