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Math Help - Someone PLEASEEE answer this T-INVARIANT QUESTION

  1. #1
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    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...
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  2. #2
    Super Member Rebesques's Avatar
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    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
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  3. #3
    Super Member Rebesques's Avatar
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    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}
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