Remark 2

Tag 081K, permanent. Cite it as [Bourbaki, Tag 081K].

Topological Vector Spaces, chapter V, § 4. Some classes of operators in hilbertian spaces

Let $M$ and $N$ be two closed vector subspaces of $E$ , and $p_M$ (resp. $p_N$ ) the orthoprojector from $E$ onto $M$ (resp. $N$ ). Then $M \subset N$ if and only if $p_M \leq p_N$ . For, we have $p_M^* p_M = p_M$ , hence

\|p_M(x)\|^2 = \langle p_M(x)|p_M(x)\rangle = \langle x|p_M^* p_M(x)\rangle = \langle x|p_M(x)\rangle

for all

x \in E

. The relation

p_M \leq p_N

is therefore equivalent to «

\|p_M(x)\| \leq \|p_N(x)\|

for all

x \in E

». If

M \subset N

, we have

p_M = p_M p_N

, hence

\|p_M(x)\| \leq \|p_N(x)\|

since

\|p_M\| \leq 1

. Conversely, if

\|p_M(x)\| \leq \|p_N(x)\|

for all

x \in E

, the kernel of

p_M

contains the kernel of

p_N

, that is, that

M^\circ \supset N^\circ

, which implies that

M \subset N

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

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Cites

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