Projektivní prostor

Projektivní prostor je geometrická a algebraická struktura.

Abstraktně se pro vektorový prostor $V$ nad komutativním tělesem $F$ definuje projektivní prostor $\mathbb {P} (V)$ jako množina všech jeho (neorientovaných) směrů (tj. jednorozměrných vektorových podprostorů):

\mathbb {P} (V):=\{Fv\,|\,v\in V\backslash \{0\}\}

resp. ekvivalentně jako množina tříd ekvivalence na množině nenulových vektorů $V\backslash \{0\}$ , pokud relaci ekvivalence definujeme jako (lineární závislost vektorů):

v\simeq w\quad \Leftrightarrow \quad v=aw

pro nějaké $a\in F$ .

Projektivní prostor $n$ -rozměrného vektorového prostoru nad tělesem $F$ se také někdy značí $\mathbb {FP} ^{n-1}$ a jeho dimenze se definuje jako ${n-1}$

Příklady

Speciálním případem projektivního prostoru je reálná projektivní rovina $\mathbb {RP} ^{2}$ , kterou dostaneme volbou $V=\mathbb {R} ^{3}$ .

Jednorozměrný komplexní projektivní prostor (komplexní projektivní přímka) $\mathbb {CP} ^{1}$ je difeomorfní dvourozměrné sféře. Jedná se dokonce o holomorfní varietu.

Fanova rovina je nejmenší projektivní rovina skládající se ze 7 bodů a 7 přímek, dostaneme ji jako $\mathbb {Z} _{2}\mathbb {P} ^{2}$ , kde $\mathbb {Z} _{2}$ je dvouprvkové těleso.

Vlastnosti

Pro reálný resp. komplexní vektorový prostor $V$ konečné dimenze má projektivní prostor $\mathbb {P} (V)$ přirozenou strukturu hladké variety. Tato varieta je kompaktní.

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