Exercise 6.3.12

Answers

1.

x \in R {(T^{∗})}^{⊥}

we have

0 = ⟨ x, T^{∗} (y) ⟩ = ⟨ T (x), y ⟩

for all $y$ . This means that $T (x) = 0$ and so $x \in N (T)$ . Conversely, if $x \in N (T)$ , we have

⟨ x, T^{∗} (y) ⟩ = ⟨ T (x), y ⟩ = 0

for all $y$ . This means that $x$ is an element in $R {(T^{∗})}^{⊥}$ .

2.

By Exercise 6.2.13(c) we have

N {(T)}^{⊥} = {(R {(T^{∗})}^{⊥})}^{⊥} = R {(T^{∗})}^{⊥} .

jephianlin

2011-06-27 00:00