In a Hilbert space $H$, given a closed subspace $M$, we know $H=M \oplus M^{\perp}$. So my question is, can we say if a vector is not in $M$, then it has to be in $M^{\perp}$?
2026-04-06 20:56:56.1775509016
A question about Hilbert space
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Take $\mathbb{R}^2$ and $M = \operatorname{sp} \{ e_1 \}$. Then $M^\bot = \operatorname{sp} \{ e_2 \}$.
Clearly there are more points than $M \cup M^\bot$.