Exercise 7.A.3

Answers

Proof. Suppose $U$ is invariant under $T$ . Let $u \in U$ and $w \in U^{⊥}$ . Then

⟨ u, T^{∗} w ⟩ = ⟨ Tu, w ⟩ = 0,

where the second equality follows because $Tu \in U$ . Thus $T^{∗} w \in U^{⊥}$ .

The converse is the same, because ${(T^{∗})}^{∗} = T$ and ${(U^{⊥})}^{⊥} = U$ . □

celiopassos

2017-10-06 00:00