Exercise 2.2.15

Answers

1.: We have zero map is an element in $S^{0}$ . And for $T, U \in S^{0}$ , we have $(T + cU) (x) = T (x) + cU (x) = 0$ if $x \in S$ .
2.: Let $T$ be an element of $S_{2}^{0}$ . We have $T (x) = 0$ if $x \in S_{1} \subset S_{2}$ and hence $T$ is an element of $S_{1}^{0}$ .
3.: Since $V_{1} + V_{2}$ contains both $V_{1}$ and $V_{2}$ , we have ${(V_{1} + V_{2})}^{0} \subset V_{1}^{0} \cap V_{2}^{0}$ by the previous exercise. To prove the converse direction, we may assume that $T \in V_{1}^{0} \cap V_{2}^{0}$ . Thus we have $T (x) = 0$ if $x \in V_{1}$ or $x \in V_{2}$ . For $z = u + v \in V_{1} + V_{2}$ with $u \in V_{1}$ and $v \in V_{2}$ , we have $T (z) = T (u) + T (v) = 0 + 0 = 0$ . So $T$ is an element of ${(V_{1} + V_{2})}^{0}$ and hence we have ${(V_{1} + V_{2})}^{0} \supset V_{1}^{0} \cap V_{2}^{0}$ .

jephianlin

2011-06-27 00:00