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BwataBaire.SousEspacesVectorielsStrictsr1.2 - 28 May 2006 - 20:44 - ThierryMonteiltopic end
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Sous espaces vectoriels stricts

Version topologique (Baire)

Ennonce

Un Banach n'est pas union denombrable de sous-espaces vectoriels fermes stricts.

Preuve

faut pas deconner

Necessite

\[ \mathbb{R}[X]=\bigcup_{n\in\mathbb{N}} \mathbb{R}_{n}[X] \]

Version algebrique (Dirichlet)

Ennonce

Si $K$ est un corps infini, un $K$-espace vectoriel n'est pas union finie de sous-espaces vectoriels stricts.

Preuve

Necessite

\[ {\mathbb{F}_{2}}^2 = \{(0,0),(0,1)\} \cup \{(0,0),(1,0)\} \cup \{(0,0),(1,1)\} \]

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