Exercise 5.4.7

Answers

Let $W$ be a $T$ -invariant subspace and $T_{W}$ be the restricted operator on $W$ . We have that

R (T_{W}) = T_{W} (W) = T (W) \subset W .

So at least it’s a well-defined mapping. And we also have

T_{W} (x) + T_{W} (y) = T (x) + T (y) = T (x + y) = T_{W} (x + y)

and

T_{W} (cx) = T (cx) = cT (x) = c T_{W} (x) .

So the restriction of $T$ on $W$ is also a linear operator.

jephianlin

2011-06-27 00:00