Exercise 6.3.17

Answers

If $x \in R {(T^{∗})}^{⊥}$ we have

0 = {⟨ x, T^{∗} (y) ⟩}_{1} = {⟨ T (x), y ⟩}_{2}

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) ⟩}_{1} = {⟨ T (x), y ⟩}_{2} = 0

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

jephianlin

2011-06-27 00:00