1. ## Someone PLEASEEE answer this T-INVARIANT QUESTION

Assume W is a subspace of v.space V and that T is a linear operator.

Suppose that V = R(T) + W (DIRECT SUM) and W is T-Invariant.

Show by example that the conclusion for the thoerem if V IS FINITE-DIMENSIONAL THEN W = N(T) is not necessarily true if V is not finite dimensional...

2. Define $V=L(0,1)$, $T(u)=\begin{cases}u_e, u\neq0\cr 1,u=0 \end{cases}\$ (denoting the even part)

and
$W=\{u: u=u_0\}$ (denoting the odd part).

Then $V=R(T)\oplus W, \ 0\notin N(T)=\{u=u_0, u\neq0\}$ but $0\in W$

3. My previous answer is SO WRONG!!

Must be careful what I say henceforth...

Try the map $T:\ell^2\rightarrow\ell^2$ defined by $T(a_n)=a_n-a_{2n}$