Proposition 6

Tag 05S5, permanent. Cite it as [Bourbaki, Tag 05S5].

If M and N are two subspaces of a vector space E, then

\operatorname{codim}(M + N) + \operatorname{codim}(M \cap N) = \operatorname{codim} M + \operatorname{codim} N.

(10) It suffices to apply Corollary 2 of Proposition 4 to the exact sequence

0 \to E/(M \cap N) \to (E/M) \oplus (E/N) \to E/(M + N) \to 0

(§ 1, no. 7, Proposition 10) and use no. 2, Proposition 2.

Note that if E is finite dimensional, (10) is a consequence of (7) and (9).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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