Discussion
Let \(H\) be a Hilbert space and \(S^{\perp}\) denote the orthogonal complement of a set \(S \subseteq H\). Which of the following is INCORRECT?
Answer: A
The incorrect statement is \(S_{1} \subseteq S_{2} \Rightarrow S_{1}^{\perp} \subseteq S_{2}^{\perp}\).
Taking orthogonal complements reverses inclusion. A vector orthogonal to every element of the larger set \(S_{2}\) is orthogonal to every element of \(S_{1}\), so the true statement is \(S_{2}^{\perp} \subseteq S_{1}^{\perp}\). For example, \(\{0\} \subseteq H\) but \(\{0\}^{\perp} = H\) is not contained in \(H^{\perp} = \{0\}\).
The other three are correct. Every \(s \in S\) is orthogonal to every element of \(S^{\perp}\), so \(S \subseteq (S^{\perp})^{\perp}\). Every vector is orthogonal to 0, so \(\{0\}^{\perp} = H\). And \(S^{\perp}\) is an intersection of kernels of the continuous functionals \(x \mapsto \langle x, s \rangle\), hence always closed.
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