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Exercise 5.A.4

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Proof. Suppose $u \in U_{1} + \dots + U_{m}$ . There are $u_{1} \in U_{1}, \dots, u_{M} \in U_{m}$ such that

u = u_{1} + \dots + u_{m}

But

Tu = T u_{1} + \dots + T u_{m} .

Because, for each $j = 1, \dots, m$ , $U_{j}$ is invariant under $T$ , it follows that $T u_{j} \in U_{j}$ . Therefore $Tu \in U_{1} + \dots + U_{m}$ . □

celiopassos

2017-10-06 00:00

Exercise 5.A.4

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